markov analysis in manpower systemchain models. markov chain models have been applied in examining...
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MARKOV ANALYSIS IN MANPOWER SYSTEM
Thesis submitted in partial fulfilment for the award of
Degree of Doctor of Philosophy in Mathematics
By
V. AMIRTHALINGAM
Under the Guidance and Supervision of
Dr. P. MOHANKUMAR M.Sc., M.Phil., Ph.D.,
VINAYAKA MISSIONS UNIVERSITY
SALEM, TAMILNADU, INDIA
July - 2016
VINAYAKA MISSIONS UNIVERSITY
DECLARATION
I, V. AMIRTHALINGAM declare that the thesis entitled
“MARKOV ANALYSIS IN MANPOWER SYSTEM” submitted by
me for the Degree of Doctor of Philosophy in Mathematics is the record
of work carried out by me during the period from
January 2011 to July 2016 under the guidance of
Dr. P. MOHANKUMAR, M.Sc., M.Phil., Ph.D., Professor, Department
of Mathematics, Aarupadai Veedu Institute of Technology, Chennai and
that has not formed the basis for the award of any other degree, diploma,
associateship, fellowship, titles in this or any other university or other
similar institutions of higher learning.
Signature of the Candidate
Place: Salem
Date:
VINAYAKA MISSIONS UNIVERSITY
CERTIFICATE BY THE GUIDE
I, Dr. P. MOHANKUMAR M.Sc., M.Phil., Ph.D., certify that the
thesis entitled “MARKOV ANALYSIS IN MANPOWER SYSTEM”
submitted for the Degree of Doctor of Philosophy in Mathematics by
V. AMIRTHALINGAM is the record of research work carried out by
his during the period from January 2011 to July 2016 under my
guidance and supervision and that this work has not formed the basis for
the award of any degree, diploma, associateship, fellowship or other titles
in this University or any other University or Institutions of higher
learning.
Signature of the Supervisor with designation
Place: Salem
Date:
ACKNOWLEDGEMENT
At first, I would like to express my deepest gratitude to my
research supervisor and guide Dr. P. Mohankumar, without whose
guidance none of this would have been possible. His encouragement,
support, freedom and keen insight have been invaluable and I have indeed
been fortunate to have such an ideal advisor. He played a very important
role in leading me towards scientific maturity. I am indebted to his
support through the years on both scientific and personal matters.
I am grateful to the founders of Vinayaka Missions University
especially I thank respected Madam Founder Chancellor, Vinayaka
Missions University Mrs. Annapoorani Shanmugasundaram,
Dato Dr. S. Sharavanan, Vice Chairman, Vinayaka Missions and
respected Chancellor Dr. A. S. Ganesan, for their valuable support.
I thank the Vice-Chancellor Prof. Dr. V. R. Rajendran, and Registrar
Prof. Dr. Y. Abraham of Vinayaka Missions University and I express
my sincere thanks to Dr. K. Rajendran, Dean (Research) Vinayaka
Missions University, Salem, Tamilnadu, India.
I express my profound thanks to Dr. A. Nagappan, Principal,
VMKV Engineering College, Salem, Tamilnadu, India. Whose constant
encouragement to carry out this research activity and for providing me
with necessary help.
5
I am thankful to Dr. M. Nithya, Head of the Department of
Computer Sciences, VMKV Engineering College, Salem and my sincere
thanks are due to Dr. S. Kandasamy, Head of the Department of Science
and Humanities and my colleagues, Professors of VMKV Engineering
College, Salem Tamilnadu, India.
I am indebted to my wife, my son and friends for their timely and
valuable support and cooperation throughout my studies.
Above all, my sincere thanks are due to the Lord ‘The Supreme’
for providing opportunity of enlightment.
V. AMIRTHALINGAM
6
CONTENTS
CHAPTERS TITLES Page No
Chapter 1 INTRODUCTION 1
1.1 Overview … 1
1.1.1 Recruitment … 8
1.1.2 Promotion … 9
1.1.3 Wastages … 12
1.2 Techniques Used in Manpower Models … 13
1.2.1 Renewal theory … 13
1.2.2 Markov renewal theory … 16
1.2.3 Semi-Markov processes … 20
1.2.4 Stochastic point processes … 21
1.2.5 Product densities … 22
1.3 Heterogeneity … 23
Chapter 2 REVIEW OF LITERATURE 29
Chapter 3 MANPOWER PLANNING PROCESS WITH
TWO GROUPS USING STATISTICAL
TECHNIQUE
61
3.1 Introduction … 61
3.2 Description and Analysis of the Model
… 62
3.3 Main Result
… 63
3.4 Numerical Illustration … 65
3.5 Conclusion … 67
7
Chapter 4 OPTIMAL MANPOWER RECRUITMENT AND PROMOTION POLICIES FOR THE TWO
GRADE SYSTEM USING DYNAMIC
PROGRAMMING APPROACH
68
4.1 Introduction … 68
4.2 Manpower System Costs … 69
4.3 Mathematical Model … 72
4.4 Dynamic Programming Formulation … 74
4.5 The Manpower Planning Horizon Theorem … 77
4.6 Numerical Illustration … 78
4.7 Conclusions … 79
Chapter 5 APPLICATIONS OF DIFFERENCE EQUATIONS 80
5.1 Introduction … 80
5.2 Model 1: Description and Analysis of the Model … 81
5.2.1 Main Results … 84
5.2.2 Special Case … 93
5.2.3 Numerical Illustration … 94
5.2.4 Conclusions … 95
5.3 Model 2: Description and Analysis of the Model … 95
5.3.1 Main Results … 98
5.3.2 Special Case … 100
5.3.3 Numerical Illustrations … 100
5.4 Conclusions … 101
8
Chapter 6 MARKOV ANALYSIS OF BUSINESS WITH
TWO LEVELS AND MANPOWER WITH
THREE LEVELS
103
6.1 Introduction … 103
6.2 Assumptions … 104
6.3 System Analysis … 105
6.4 Numerical Illustration … 109
6.5 Conclusion … 110
Chapter 7 EXPECTED TIME FOR RECRUITMENT IN A
TWO GRADEDMANPOWER SYSTEM
ASSOCIATED WITH CORRELATEDINTER-
DECISION TIMES WHEN
THRESHOLDDISTRIBUTION HAS SCBZ
PROPERTY
113
7.1 Introduction … 113
7.2 Model 1: Description and Analysis of the Model … 114
7.2.1 Main Results … 117
7.2.2 Special Case … 122
7.2.3 Numerical Illustration … 123
7.2.4 Conclusion … 124
7.3 Model 2: Description and Analysis of the Model … 124
7.3.1 Main Results … 126
7.3.2 Special Case … 128
7.4 Conclusion … 129
References … 130
Publications … 142
1
CHAPTER 1
INTRODUCTION
1.1 OVERVIEW
The analyses of manpower systems have become very important
component of planned economic development of any organization or
nation. However, manpower planning depends on the highly
unpredictable human behavior and the uncertain social environment in
which the system functions. Hence the study of probabilistic or stochastic
models of manpower systems is very much essential. Several stochastic
models of manpower systems have been proposed and studied
extensively in the past (see Bartholomew (1967) and Vajda (1978)).
Various stochastic models of manpower systems can be classified broadly
into two types:
1) Markov Chain models
2) Renewal Models
In all these models, the manpower system is hierarchically graded
into mutually exclusive and exhaustive grades so that each member of the
system may be in one and only one grade at any given time. These grades
are defined in terms of any relevant state variables. Individuals move
between these grades due to promotions or demotions and to the outside
world due to dissatisfaction, retirement or medical reasons. If the size of
2
the grades is not fixed, then the state of the system at any time is
represented by a vector 2 ,( ,l nX t X t X t X t where the
component t represents the number in the thi grade at any time t .
Further the very nature of several manpower systems require to be
observed at, say, annual intervals. Accordingly, the system behaviour is
adequately described by a Markov chain, such models are called Markov
chain models.
Markov chain models have been applied in examining the structure
of manpower systems in terms of the proportion of staff in each grade or
age profile of staff under a variety of conditions and evaluating policies
for controlling manpower systems (see for example, Young and Almond
(1961), Young (1971), Forbes (1971a,b), Bartholomew (1973) and Gani
(1973)). In these works and in all of what followed the important question
was the control of the expected numbers in the various states by
recruitment control. The numbers of people in such categories change
over time through wastage, promotion flows and recruitment. Some of
these flows are subject to management control while others vary in a
random manner. Factors such as the need to offer adequate career
prospects or the requirement of the job will often dictate a desirable age
or grade structure and it is the manpower planner's task to determine
whether this can be achieved and , if so, how.
3
The limiting behavior of an expanding non-homogeneous Markov
system has practical importance as shown by the literature on manpower
systems (Vassiliou 1981a&b, 1982a). The limiting structure of the
expected class sizes was derived under certain conditions and the relative
limiting structure is shown to exist with a different set of conditions.
Mehlmann (1977) and Vassiliou (1982b) studied the limiting behavior of
the system with Poisson recruitment and observed that the number in the
various grades are asymptotically mutually independent Poisson.
Vassiliou (1984c) studied the asymptotic behavior of non-homogeneous
Markov systems under the cyclical behavior assumption and provided a
general theorem for the limiting structure of such systems. Vassiliou
(1986) later extended the results and provided a basic theorem for the
existence and determination of the limiting structure for the vector of
means, variances and covariances under more general possible
assumptions. He argued that the results are useful from the practical point
of view since they provide valuable information about the inherent
tendencies in the system.
The control of asymptotic variability of expectations, variances and
covariances in a Markov chain model is a major research area in
manpower systems. The earliest work on this subject was that of Pollard
(1966). The results were later extended by several authors (Vassiliou and
Gerontidis (1985), Vassiliou (1986), Vassiliou et al. (1990)). Attainable
4
and maintainable structures in Markov manpower systems under
recruitment control have been studied by Bartholomew (1977), Davies
(1975, 1982), Vassiliou and Tsantas (1984 a&b) and later Kalamatianou
(1987) analysed the same with pressure in grades. The concept of a non-
homogeneous Markov system in a stochastic environment (S-NHMS)
was introduced for the first time by Tsantas and Vassiliou (1993). The
problem of attaining the desired structure in an optimal way as well as
maintaining relative grade sizes applying recruitment control in a
stochastic environment as introduced in Bartholomew (1975, 1977) is
considered. More references in this and related topics an be found in
various papers by (Georgiou (1992), Tsantas (1995), Tsantas and
Georgiou (1994, 1998)). A Markov model responding to promotion
blockages has been proposed by Kalamatianou (1988). Raghavendra
(1991) has employed a Markov chain model in obtaining the transition
probabilities for promotion in a bivariate framework consisting of
seniority and performance rating. Georgiou and Vassiliou (1997) have
introduced phases in a Markov chain model and investigated the input
policies subject to cost objective functions. Yadavalli and Natarajan
(2001) studied a semi-Markov model in which a single grade system
allows for wastage and recruitment. The time dependent behaviour of
stochastic models of manpower system with the impact of pressure on
promotion was subsequently studied by Yadavalli et al. (2002).
5
Although a Markov model is simple and easy to implement, it does
not take into account existing knowledge of the distribution of length of
service until leaving. In such cases the mathematically intractable Semi-
Markov models approach is suggested (McClean 1991). The Semi-
Markov processes are a generalization of Markov processes in which the
probability of leaving a state at a given point in time may depend on the
length of time the state has been occupied (duration of stay) and on the
next state entered. However, there are several theoretical literatures on
Semi-Markov Models ( Pyke (1961 a & b), Ginsberg (1971), Mehlmann
(1979), McClean (1978, 1980, 1986)). A stochastic model of migration,
occupational and vertical mobility, based on the theory of Semi-Markov
process was derived by Ginsberg (1971). McClean (1978) extended the
assumption of simple Markov transitions between grades and the leaving
process to semi-Markov formulation which allows for inclusion of well-
authenticated leaving distributions such as the mixed exponential.
Moreover, the previous assumption of Poisson recruitment is generalized
to allow for a recruitment process which may vary with time, either as a
mixed exponential time dependent Poisson process or by assuming that
the number of recruits depends on the amount of capital owned by the
firm. The previous formulation is therefore extended to take into account
the fact that recruitment to a firm is a highly variable process and the
assumption of Poisson recruitment to each grade is therefore restrictive.
6
The concept of non-homogeneous semi-Markov systems found
important applications in manpower system particularly in the subjects of
variability, limiting distributions and maintainability of grade sizes
(Vasiliou and Papadopoulou (1992)).
On the other hand, there are several manpower systems where the
grade sizes are fixed by the budget or amount of work to be done.
Recruitment and promotion can occur only when vacancies arise through
leaving or expansion. There may be randomness in the method by which
vacancies are filled. The movements of individuals are characterised by
replacements (renewals) according to some probabilistic law, and such
models of manpower systems are called renewal models. The main
advantage of these models over the Markov chain models is that they are
closer to reality since the losses (wastages) occur continuously in time
and there is always the possibility that a new recruit may also leave
during the study period. White (1970) has used models of this kind to
study the flows of clergy of several large American denomination.
Stewmann (1975) has applied White's methods to the study of
recruitment and losses in a state police force. Bartholomew (1982) has
provided a detailed analysis of renewal models of manpower systems.
Sirvanci (1984) has applied renewal processes to forecast the manpower
losses of an organisation in order to determine whether the organisation
will be able to meet its demand for manpower under present conditions.
7
The distributions of completed length of service (CLS) in these
models have been fitted to actual data from industry by several
researchers (see Bartholomew, 1982). McClean (1976, 1978) has used a
mixed exponential distribution for CLS and estimated the parameters
using data for two companies. Agrafiotis (1983, 1984, and 1991) studied
the problem of labour turnover by using renewal process type models.
A satisfactory model of manpower system should provide answers to
the following questions:
1. How to provide estimates of manpower indicators of the system?
2. How to predict the future behaviour of the system under various
assumptions?
3. How to find optimum solutions to various policy problems subject
to various constraints given by the management?
4. How to avoid various problems by giving a warning before the
situation develops?
5. How to design manpower, which is related to various problems of
prediction in consultation with management?
In order to provide answers to questions raised above, the model
considered should incorporate the following main factors, which
predominantly determine the behaviour of a manpower system:
1. Recruitment
2. Promotion of employees
3. Wastages.
8
1.1.1 Recruitment
The sizes of various grades, which respond to the expansion,
promotions and wastages, are maintained at the desired level at any time
by a process called recruitment. The flow of recruitment can be
controlled by the management authorities. The recruitment can be made
in several ways. Vacancies can be filled as and when they arise or they
may be allowed to accumulate and then filled up at specified periods or
whenever the total number of vacancies attains a certain specific level, so
as to minimize the cost. The recruitment can be made by the organization
itself or by some external agencies to avoid delay and huge overhead
costs. Several organizations in South Africa do not recruit employees by
themselves (e.g. the preliminary process of senior level positions in
Statistics South Africa) but approach recognized recruiting agencies.
Usually, vacancies that arise are allowed to accumulate for a specified
period of time, or to attain a specified level and then these agencies are
requested to fill them up and to complete the process of recruitment in a
specified period of time. However, they may not be able to fill up all the
notified vacancies due to the non-availability of suitable candidates
with prescribed qualifications and experience. Further additional
vacancies may also arise during the period of recruitment process.
Therefore there may exist some vacancies even after the process of
recruitment is completed. In reality, many such manpower systems exist.
9
However, these types of models have not been considered in the
literature. Davies (1975) considered a fixed size Markov chain model that
suffered losses and admits recruits to various grades in such a manner that
the total grades in the system remain constant. In that paper, the
recruitments take place at integral points in time and at the time of
recruitment, no vacancy is left unfilled. Vassiliou et al. (1990) deal with a
non-homogeneous Markov manpower system, which allows recruitment
in each grade of the hierarchically graded manpower system. They have
obtained the limiting expected structure of the system by control over the
limit of the recruitment probabilities. Rao (1990) has considered a
manpower planning model with the objective of minimizing the
manpower cost with optimal recruitment policies. The recruitment size is
known and fixed in this model. Hence the study of a model where
vacancies are accumulated and then filled up deserves attention.
1.1.2 Promotion
Normally vacancies that arise in the lower grade are filled up by
recruitments where as those in the higher grades are filled up by
promotions. Further, promotions besides giving due recognition to
proficiency and credibility of the employees reduce the chance of an
efficient employee leaving the organization. Some of the promotion rules
are given below:
10
i The senior most in the grade is promoted.
ii Promotion is given at random.
iii Those who fill certain efficiency criterion along with some
minimum completed length of service are promoted.
As per the rule (i), the length of service is the sole criterion for
promotion and hence the management can control it. The rule (ii) gives
full freedom for the management to promote any employee of their
choice, which also is not desirable. Normally rule (iii) is preferred. Some
of the reasons, which influence the promotion policies, are (a) pressure
(b) efficiency and (c) length of service.
(a) Pressure
In a multi-graded hierarchical manpower system, a promotion
policy that is associated with constant promotion probabilities leaves a
proportion of employees qualified by completed length of service in a
lower grade un-promoted. This proportion increases and pressure starts
building up as time progresses. When pressure exceeds a certain level of
control, a high proportion of un-promoted employees could have serious
effect on the efficiency of the organization for several reasons such as
productive loss and wastage. The pressure can be quantified as a function
of the proportion of the people in a job grade according to Kalamatianou
(1987, 1988). She has quantified pressure in three stages and suggested
11
models to reduce the pressure by suitably changing the promotion
policies well in advance.
(b) Efficiency (training)
Training of manpower has long been recognized as an important
factor for improving the efficiency of the employees and for the
productive improvement. Further, when it is considered as a criterion for
promotion, it becomes very much effective. Mathematical models
incorporating training aspects have been studied by Guardabassi et al.
(1969), Grinold and Marshall (1977), Mehlmann(1980) and Vajda
(1978). Goh et al. (1987) have analysed the training problem within an
organisation using dynamic programming principles. These results were
recently generalised using Dynamic Programming by Yadavalli et al.
(2002).
(c) Length of service
Length of service in a grade should necessarily be a natural
criterion for promotion in order to create a healthy atmosphere among the
employees. However, for controlling the promotion, the management can
include other efficiency criterion along with it for promotion. This aspect
has been discussed by Bartholomew (1973, 1982), Glen (1977) and in the
thesis of Kamatianou (1983).
12
1.1.3 Wastages
When employees move from one grade to another, they are
exposed to different factors influencing them to leave the organization.
Various data indicate that the reasons for leaving can be classified into
the following cases:
(i) Discharge
(ii) Resignation
(iii) Redundancy
(iv) Retirement
(v) Medical retirement
Agrafotis (1984) has grouped the above cases into two main
reasons, normally, (a) unnatural and (b) natural. Unnatural reasons for
leaving depend on the internal structure of the company or organisation,
viz, lack of promotion prospects, job satisfaction, problem of adjustment,
etc., including the cases (i), (ii), and (iii) mentioned above. Natural
reasons for leaving the organisation do not depend on the internal
structure of the organisation, including the cases under (iv) and (v). In
analysing data on a number of companies, Agrafiotis (1984) has shown
that there is a significant difference in the wastage rates corresponding to
reasons (a) and (b) for leaving. However, the cases (iv) and (v) relating to
the natural leaving are entirely different and are to be discussed
separately, for an employee leaving by way of natural retirement after
having served the organisation completely cannot be grouped with an
13
employee who leaves the organisation by way of medical reasons. As
such, there are three different wastage rates:
(a) Due to internal structure
(b) Due to retirement
(c) Due to medical reasons
Unlike natural wastage the unnatural wastage can be controlled by
the management by resorting to better promotional prospects, improved
working conditions and training.
Some other manpower studies which investigated wastage
intensities are (Vassiliou (1976, 1982), Leeson (1981, 1982), McClean et
al. (1992)).
1.2 TECHNIQUES USED IN MANPOWER MODELS
Here, we present the various techniques used in the analysis of
models of manpower systems.
1.2.1 Renewal theory
Renewal theory forms an important constituent in the study of
stochastic processes and is extremely used in the analysis of manpower
models with recruitment. Feller (1941, 1968) made significant
contributions to renewal theory giving the proper lead. Smith (1958) gave
an extensive review and highlighted the applications of renewal theory to
a variety of problems. A lucid account of renewal theory is given by Cox
(1962).
14
Definition 1
Let : 1,2.,,,iX i be a collection of random variables, which are
non-negative, independent and identically distributed. Then the sequence
nX is called a renewal process. We assume that each of the random
variable iX has a finite mean . A renewal process is completely
determined by means of .f , the p.d.f of iX . Associated with the
renewal process is a random variable ,N t which represents the number
of renewals in the time interval (0, ].t N t is also known as the renewal
counting process (Parzen, 1962).
Definition 2
The expected value of N t is called the renewal function and is
denoted by H t . The derivative of H t if it exists, is denoted by h t
and is called the renewal density. The quantity h t dt has the
interpretation that it represents the probability that a renewal occurs in
, .t t dt We will have to identify this as what is known as the first order
product density for a more general process. The renewal density satisfies
the following integral equation:
0
t
h t f t f u h t u du
One of the important and useful theorems in application is the key-
renewal theorem (Smith, 1958).
15
Theorem
Let Q i satisfy the following conditions:
(i) 0Q t for all 0t
(ii) Q t is non-increasing
(iii) 0
Q t dt
Then,
0
limtQ Q t u dH u
0
1Q u du
Further details regarding renewal theory can be found in Smith
(1958), Feller (1968), Prabhu (1965) and Srinivasan (1974). We now
briefly indicate how renewal theory has been used in the study of
manpower models. The stochastic element in manpower systems occur
principally due to the loss mechanism arising out of staff moving out of
the system. The randomness may also be due to the method by which the
vacancies are filled. In the context of manpower planning, the renewal
process , 0N t t represents the number of recruitments required for
the given position for which the first person was employed at 0.t The
random time X between successive replacements is called the completed
length of service (CLS) and its distribution F x is termed as the CLS
distribution. Thus, during the operation period from 0t up to time ,t
while N t employees leave, an equal number need to be recruited in
order to keep a given position continuously staffed. To predict the value
16
of N t for any given time, its expected value, which is referred to as the
renewal function, may be used. The relationship between the CLS
distribution and the renewal density ,h t the derivative of ,H t is given
by the renewal equation
0
; 0.t
h t f t f u h t u du t
Where f t is the density of the CLS distribution .F t The
renewal density h t can be interpreted as the rate at which the losses
occur. On the other hand, .F t is the distribution of the time an
employee spends in the organisation before leaving. The renewal process
of personnel losses has been extensively studied by Bartholomew (1962,
1982) and Bartholomew and Forbes (1979).
1.2.2 Markov renewal theory
Let E be a finite set, N the set of non-negative integers and
[0, ).R Suppose we have, on a probability space , ,B P random
variables : , :n nX E T R defined for each n N so that
0 1 20 ....T T T
Definition 1
The stochastic process , , ;n nX T X T n N is said to be a
Markov renewal process with the state space E provided that
1 1 0 1 0 1, \ , ,... ; , ,...n n n n nP X j T T t X X X T T T
17
1 1, /n n n n nP X j T T t X
for all ,n N j E and .t R
We assume that ,X T is time-homogeneous, that is, for any
,i j E and t R
1 1, , , /n n n nQ i j t P X j T T t X i
independent of n. The family of probabilities
, , ; ,Q i j t i j E t R
is called a semi-Markov kernel over E. We assume that
, ,0 0Q i j for all , .i j E
For each pair ,i j the function , ,t Q i j t has all the properties
of a distribution function except that;
, , ,tP i j Q i j t
is not necessarily 1. It is easy to see that
, 0,P i j , 1j E
P i j
that is, ,P i j are the transition probabilities for some Markov chain
with state space E. It follows from the definition 1 and above that
1 0 1 0 1 \ , ; , .. ). (n n n nP X j X X X T T T P X j
for all ,n N j E
This implies that ;nX X n N is a Markov chain with state
space E and the transition matrix P.
18
1.2.2.1 Markov Renewal Functions
We write 1P A for the conditional probability 0[ ]/P A X i and
similarly iE for the conditional expectations given 0 .X i We also
assume that
0 1 2 0 0.P T T T
Let us define , ,nQ i j t as
, ,nQ i j t , ;n nP X j T t , ,i j E t R for all .n N
Then,
0 , ,Q i j t1
0ij
if i j
if i j
for all 0t and 0n
where ij is the Kronecker delta function.
We have the recursive relation
1 , ,nQ i j t 1
0, , , , |n
j E
Q i j ds Q j k t s
Where the integration is over [0,0) The expression , ,R i j t that
gives the expected number of renewals of the position j in the interval
[0,0) is given by
, ,R i j t 0
, ,n
n
n
Q i j t
This is finite for any ,i j E and .t The , ,R i j t are called
Markov renewal functions and the collection
19
, , ; , ,R R i j t i j E t R of these functions is called the
Markov renewal kernel corresponding to Q. We note that for fixed
,i j E the function , , ,i j R i j t is a renewal function. We can now
easily see from the various expressions above that 1
,aR I Q
where
1 is the unit matrix.
1.2.2.2 Markov Renewal Equations
The class of functions B which we will be working with is the set
of all functions
:f E R R
such that for every i E the function ,t f i t is Borel measurable and
EXR bounded over finite intervals and for every fixed j E the
functions , , ,ni j Q i j t and , , , , , ,i j R i j t i j R i j t is both
belong to B. For any function .f B the function Q f defined by
,Q f i t 1
0, , ,n
j E
Q i j ds f j t s
is well defined and Q f B again. Hence the operation can be repeated,
and the thn iterate is given by
,Q f i t 1
0, , ,n
j E
Q i j ds f j t s
We can replace Q by R, which is again a well-defined function,
which we will denote by ,R f that is for ,f B
20
R f 1
0, , , .
j E
R i j ds f j t s
A function f B is said to satisfy a Markov renewal equation if
for all i B and ,t R
,f i t 1
0, , , ,
j E
g i t Q i j ds f j t s
for some function g B
Limiting ourselves to functions ,f g B which are non-negative
and denoting this by ,B the Markov renewal equation now becomes
, ,f g Q f f g B
This Markov renewal equation has a solution .R g Every solution
f is of the form ,R f h where h satisfies , .h Q h h B For a more
detailed on Mark renewal equations see Cinclar (1975).
1.2.3 Semi-Markov processes
Let ,X T be a Markov renewal process with state space E and
semi-Markov kernel Q. Define .Sup
n nL T Then L is the lifetime of
, .X T If E is finite or if X is irreducible and recurrent, then L
almost surely. By weeding out those and t R for which
Sup
n nT we assume that Sup
n nT for all .
Then for any and t R there is some integer n N such
that 1 .n nT t T w We can therefore define a continuous time
21
parameter t t RY Y
with state space E by putting t nY X on
1.n nT t T The process t t RY Y
so defined is called a semi-Markov
process with state space E and a semi-Markov transition kernel
, , .Q Q i j t
1.2.4 Stochastic point processes
Stochastic point processes form a class of random process more
general than those considered in the previous sections. Since point
processes have been studied by many researchers with varying
backgrounds, there have been several definitions of them each appearing
quite natural from the view point of the particular problem under study
(see, for example, Bartlett (1966), Bhaba (1950), Harris (1963) and
Khinchine (1955)). A stochastic process is the mathematical abstraction,
which arises from considering such phenomena as a randomly located
population or a sequence of events in time. Typically, there is envisaged a
state space X and a set of points Xn from X representing the locations of
the different members of the population or the times at which the events
occur. Because a realization (or a sample path) of any of these
phenomena is just a set of points in time or space, a family of such
realizations has come to be called point processes (see Daley and Vere-
Jones, (1971)).
22
A comprehensive definition of a point process is due to Moyal
(1962) who deals with such process in a general space, which is not
necessarily Euclidean. Consider a set of objects each of whom is
described by a point x of a fixed set of points X. Such a collection of
objects, which we may call a population, may be stochastic if there exists
a well-defined probability distribution P on some field B of subsets of
the space of all states. We shall assume that the members of the
population are indistinguishable from one another. The state of the
population is defined as an unordered set 1 2, ,....,n
nX x x x representing
the situation where the population has n members with one of the states
1 2, ,...., nx x x . Thus the population state space is the collection of all
such nX with 0,1,2,....n where 0X denotes the empty population. A
point process is defined to be the triplet , ,B P . For a detailed treatment
of stochastic point processes with special reference to its applications the
reader is referred to Srinivasan (1974). A point process is called a regular
point process if the probability of occurrence of more than one event in
0, so .
1.2.5. Product densities
One of the ways of characterizing a general point process is
through product densities (Ramakrishnan (1950), Srinivasan (1974)).
These densities are analogous to the renewal density in the case of non-
23
renewal processes. Let ,t x denote the random variable representing
the number of events in the interval , ,t t x ,dxN t x the events in the
interval ,t x t x dx and , , , .P n t x P N t x n
The product density of order n is defined as
1 2, ,...,n nh x x x
1 2, ,... 01 2
, 1; 1,2,...lim
....n
i i
n
P N x i n
Where 1 2 .... ,nx x x or equivalently for a regular process
1 2, .....,h n x x x n
lim 1, 2,.... 0n
1 , 1; 1,2,...i n N x i i i n
where 1 2 .... .nx x x
These densities represent the probability of an event in each of the
intervals 1 1 1 2 2 2, , , ,... , .n n nx x x x x x x x x Even though the
functions 1 2, ,...n nh x x x are called densities it is important to note that
their integration will not give probabilities but will yield the factorial
moments. The ordinary moments can be obtained by relaxing the
condition that all the '
ix s are different.
1.3 HETEROGENEITY
The validity of the models described under section 1.2 depends
highly on the assumption that the manpower study be based on
homogeneous groups of individuals. This is a huge task, which can never
24
be attained in practice because human behaviour is highly unpredictable
and the environment on which the system operates is uncertain. However,
it is paramount that the researcher ensures that there is no major source of
heterogeneity. Individuals" differences depend on many factors such as
their motivation, performance and commitment to the employer.
The subject of homogeneity of individuals is fundamental in
virtually all fields of study. However, in the biomedical literature, it is a
well known fact that individuals differ substantially in their endowment
for longevity (see Manton (1981); Keyfitz (1978); Shepard and
Zeckhauser (1977). Hence it is important to try and understand the
impact of heterogeneity on the study results. In demography and public
policy analysis studies, it has been found that ignoring heterogeneity in
frailty results in biased results (Vaupel et al. (1979, 1985)).
According to Bartholomew et al. (1991) the analysis of individual
differences is of fundamental importance in the study of manpower
system, in particular, wastages (losses from the system). Any attempt to
describe wastage pattern must reckon with the fact that an individual's
propensity to leave a job depends on a great many factors, both personal
and environmental. Failure to recognise the effects of heterogeneity may
not only result in erroneous results when applying manpower models but
also complicate both the theoretical and empirical research due to the
composition of the population and the differential impact of economic,
25
environmental and social forces. The flow of people in manpower
systems, moving employees in various states can be subdivided into
recruitment stream, the transition between the state and the outflow from
the system. Considering a discreet time 0,1,...t we assume that the
individuals' transitions between the states take place either according to a
homogeneous Markov chain. Most of the work was based on
homogeneous Markov chain model introduced by Young and Almond
(1961), Gani (1963), Young (1971), and Sales (1971). Later on Young
and Vassiliou (1974), Vassiliou (1976, 1978) introduced the non-
homogeneous Markov chain model, which was reported by many
researchers to provide a good prediction in practice. Vassiliou (1982a)
introduced the more general framework of non-homogeneous Markov
model, which incorporates a great variety of applied probability models.
As the literature shows, the theory of non-homogeneous Markov systems
(NHMS) has flourished since then (Vassiliou, et al. (1990); Tsantas and
Vassiliou (1993); Georgiou (1992); Tsantas (1995))
A number of authors suggested tackling the problem of
heterogeneity by dividing the personnel system into more homogeneous
subsystems. The pioneering work on mover stayer models of labor
mobility by Blumen et al. (1955), Goodman (1961) and later
Bartholomew (1982) was one form of subdividing the population into
categories-the 'stayers' who hardly change their jobs and the 'movers'
26
who tend to change jobs frequently. Ugwuowo and McClean (2000)
proposed some techniques to deal with heterogeneity for modeling
wastage, though the problem exits in other flows within the personnel
system. To incorporate population heterogeneity into manpower
modeling, two strategies have been suggested: the use of observable
sources of heterogeneity as it affects wastage and the latent source of
heterogeneity that are impossible to observe but are known to affects the
key parameters of the model. Although the division of individuals in
homogeneous subcategories is a fundamental and important step in
application of the manpower planning techniques, there is still lack of
attention towards the way homogeneous groups can be attained in
practice. De Feyer (1996) presented a general framework to get more
homogeneous subgroups for using Markov Chain theory in manpower
planning. A general splitting-up approach is suggested as well as the use
of some statistical multivariate techniques is proposed to support the
splitting-up process.
Sathiyamoorthi and Elangovan (1998) have obtained the mean and
varianceof time to recruitment using shock model approach when (i)
loss of manpower is a non-negative integer valued random variable, (ii)
threshold for loss of manpower is a discrete random variable following
geometric distribution and (iii) the time between three consecutive
27
decision form a sequence of independent and identically distributed
random variables.
Mariappan and Srinivasan (2001,2002) have obtained the mean
and variance of the time for the recruitment using shock model approach
when (i) staff depletion are caused by the decision making epochs and the
inter - arrival time between consecutive decisions are exchangeable and
constantly correlated exponential random variables (ii) the sequences
associated with the cumulative loss of man hours due to the exodus of
personnel and the inter- decision times taken by the organization, form a correlated pair of renewal sequences. Sathyamoorthy and Parthasrathy (2002) have obtained the
expected time for recruitment when (i) loss of manpower is a
continuous random variable (ii) threshold for loss of manpower is a
continuous random variable having SCBZ property instead of
exponential distribution which has lack of memory property and the inter
decision times form a sequence of independent and identically distributed
random variables.
Saavithri and Srinivasan (2001) have obtained the expected
time for recruitment using some univariate policies of recruitment when
(i) the loss of man hours for each decision taken form a sequence
of independent and identically distributed random variables, with state
space (0,∞), (iv) survival time process and loss of man hours process are
independent with state space (0,∞).
28
Kasturri and Srinivasan (2007) have obtained the mean time to
recruitment for both the models when (i) the thresholds grades are
exponential random variables and (ii) the inter decision timeshare
exchangeable and constantly correlated exponential random
variables. Recently Kasturri and Srinivasan (2007) have
extended their results by assuming that the threshold distributions for the
two grades have SCBZ property.
29
CHAPTER 2
REVIEW OF LITERATURE
2 REVIEW OF LITERATURE
Manpower has to be wisely exploited for the steadfast growth of a
economy. This is the reason why there is Ministry Of Human Resources,
the aim is to implement plans to utilize the human resources available
through out the country for their growth and country's Development. This
is given as much an importance as any other discipline as economics,
psychology, law and public administration, industrial relations, computer
science and operations research. All the disciplines stated above are
themselves in a tremendous state of flux. Manpower planning requires a
keen study, this has necessitated the coming up of lot of literature. New
ways and means are suggested for optimum usage of manpower through
Economics, Operations Research and Mathematical Models. Research is
going on in every field for their growth and manpower planning does not
lag behind.
Manpower planning is historically rooted in the gathering of
manpower statistics dating from the times of the Roman census to the
accounting of slaves, and eventually to population census towards the
end of the eighteenth century Morton [56].
30
Historically, origin of the models of manpower systems could be
traced back to Seal [72]. However, simple models have been reported by
Edwards [20] to have been used by manpower planners long before then.
Mehlmann [54] has developed an optimal recruitment and transition
strategies for manpower systems using dynamic programming recursion
with the objective of minimizing a quadratic penalty function which
reflects the importance of correct manning of each grade under preferred
recruitment and transition patterns.
Lane and Andrew [46] has developed a lognormal model in which
the distribution of wastage was related to length of services and proposed
two methods of analysis. Cohort analysis, in which the wastage
characteristics of an initial homogeneous groups are observed over longer
periods of time; census analysis in which two sample points in time are
used to determine the wastage rates.
As alternative, approach to manpower planning is based on
optimization theory; the theoretical foundations of the optimization
approach have been developed in Holt et al. [35]. Holt developed a cost
model that includes both the costs of maintaining and changing the work
force. Holt uses a quadratic cost model that allows closed form solution
to be developed and finds that optimal staffing levels are based on the
weighted values of forecast demand.
31
A general description of the models and the methods for
developing mathematical models for manpower studies have been
discussed in a broad way by Dill et.al [19]. In this paper the authors have
explored some results in manpower and the development of some simple
stochastic models to take note of such issues in manpower planning.
Direct mathematical methods have been used for structuring such models
and the simulation methods have been used to a large extent in such
models.
Morton [56] presents a concise historical summary of forecasting
techniques starting from demographers' modified exponentials through to
renewal theory, stochastic processes, moving average ad exponential
smoothing. In recent years there has been a swing away from
demographic ally based forecasts towards econometric and input / output
models as well as Monte Carlo simulation. Moreover, recognition that
long lead time in manpower development makes planning particularly
vulnerable to changes in a policy variables. It has simulated research into
'Ideological" or target - related forecasting in which the study of
explicitly stated achievable future goals are undertaken through futurist
speculation or expert consensus in order to restrict the range of the
exogenous variables.
32
Walker [90] compresses forecasting, inventory, determining
qualitative and quantitative, recruitment, selection, training,
development, motivation and compensation into two constituents. The
determination of organizational needs and available manpower supply
within the organization at various times through forecasting; and
programming, the planning of activities which will result in the
recruitment of new employees for the organization, further development
activities for employees, designation of replacement for key managers,
and new expectations for effective top managements planning,
The author goes on to integrate the two basic elements in a time
frame short range (0-2) intermediate range (2-5 years), long range (5-10
years).
Considering recruitment and promotion as some of main activities
of the organization, Vajda [84] has discussed a very systematic account of
the applications of mathematical models to the problems of manpower. In
any organization the employees can be partitioned into different grades.
So, a population of employees in an organization has a well defined
structure in terms of the partition. The question is to find out the changes
in a given structure and how it gives rise to another structure after one or
more transitions. The second question is from a given structure in how
many steps a specified structure could be attained. The problem of
33
interest here is to find out in how many transitional steps the structure
which is identical with the starting could be attained. It implies the revisit
to the starting state in the terminology of stochastic processes.
An optimum manpower utilization model using mathematical
programming has been discussed by Schneider and Kilpatrick [71] for
health maintenance organization. The interaction between effective
manpower utilization, faculty requirements and available capital is
discussed in two basic models, one is an overall planning model and the
other is a subscriber maximization model. The objective used in these
models pertains to either minimum cost or minimum feasible use of
physicians through the substitution of physician extenders.
Girnold [30] has examined the problem of producing a commodity
with uncertain future demand with time lags in the production process
and with the commodity itself being a vital input in the production
process. Kurosu [45] research which is of relevance to job shops
situation described the influence of demand uncertainties on waiting
time, idle time and rate of losing customers. The study, modeled demand
as a queuing process and gone as far as prescribing timing and conditions
for temporary increasing or decreasing process capacity to absorb
fluctuations in demand but, however failed to consider the manpower
34
costs. Aderoba [2] has established a procedure for determining
appropriate levels of full time labour and over time engagement.
A mathematical model of a military manpower system with a view
to determine the optimal steady state, wage rate and force distribution by
length of service is by Jaquette and Nelson [39]. In this paper it is
assumed that the cost of hiring personnel is determined by military
manpower supply function which relates enlistment and re-enlistment
rates to military pay. The optimal force is defined as that force which
provides the greatest military capability for a given budget cost. Optimal
rates of pay are determined by maximizing the productivity index subject
to a budget constraint. Assuming the basic flow process as Markovian the
optimal rates of pay are determined. The steady state optimal policy for
the Cobb-Douglas type function is obtained using the Lagrangian
multiplier technique. Numerical results are also discussed.
The use of linear programming methods to derive optimal long
term policy has been discussed by Grinold and Marshall [31]. They have
introduced the long term planning horizon and uncertain conditions
pertaining to future manpower requirement. The input data as regarding
future requirements, budget constraints, cost, discount rate, utilization
factor and coefficient factor relating to How processes. The inflow is
taken as decision variable, the objective function being the discount cost.
35
The minimization of the same is discussed. A person leaves the rth
grade with probability pr. The length of time x to stay in grade r has a
probability density function rg x and a survival function rG x , in
case he leaves the grade .rS In case the person is promoted the
corresponding probability density function of the length of time to stay
which is y is f y with the corresponding survivor function .rF y
Under these assumptions the semi-Markov transitions between grades are
discussed. The mean grade size at time ' 't is obtained.
A particular aspect has received much attention in the examination
of movement structure of the state of these systems in terms of the
proportion of staff in each grade; and the evaluation of recruitment and
promotion policies for controlling them. Morgan [55], Vassilou [86] and
Lesson [48, 49] in their works have analyzed graded structures with
grades depending on promotion probabilities. The length of service is
considered as an important criterion in determining the staff flow. The
organizations such as civil service where large number of manpower is
required, the grades are sub divided in to several categories for
administrative purposes.
An interesting paper by Abounde and Mcclean [1] contains the
discussions about the model where a manpower system with a constant
level of recruitment is considered, It is related to the production planning
36
in the development of telephone services and linking the same to the
work force. The constant level of recruitment necessary to bring the
number of installations eventually up to their final is discussed. Also a
stochastic model is developed which evaluates the effect of implementing
the recruitment policies in term of changing distributions of staff
numbers, and the changing number of installations with time. Numerical
results are also provided.
Zanakis and Maret [94] have discussed a Markov chain model to
describe the manpower supply planning. In doing so, the two different
aspects of manpower planning namely the demand for manpower and
supply of manpower are considered. The authors have developed a
model, which indicates the How of personnel through an organization as
Markov chain. The authors described the stage interval like week, year,
ete, to define the time interval for transitions. Also the authors indicate
the need for the selection of exhaustive and non-overlapping stage to
which an employee can be classified. Using this information the method
of constructing the TPM is indicated. The authors have indicated the use
of the statistical procedures for testing the Markov chain assumptions.
The use of Chi-Square test for verifying the stationary and the first order
process in non-overlapping chains is indicated. The authors indicate the
method of obtaining the probability of an employee remaining in the
37
same grade of a specific length of time. The authors indicate that the
Markov chain model provides valuable insight into predicting future
organization manpower losses. Numerical examples are also worked out
to indicate the usefulness of the results.
Barthlomew [8] has discussed the form of subdividing the
population into categories, the 'stayers' who hardly change their jobs and
'movers' who tend to change jobs frequently, while Barthlomew and
Forbes [11] have developed a more specific application of the
principles, to the manpower planning problem. A basic model defines a
number of discrete manpower grades, with employees advancing or
leaving with fixed transition probabilities. The state of the system is
defined as the number of employees in each grade and the system is
analysed as a Markov chain.
Lesson [49] has considered the recruitment policies and their
effects on internal structures. Recruitment control refers to an effective
control of recruitment policies to obtain an optimal supply of manpower
for a system at any time. Generally recruitment levels are connected with
wastage and promotions in a system as well as the desired growth of the
system, hence controlling recruitment policies may help to attain a
desired structure, which could be maintained over a time.
38
The paper by Agrafotis [3] is on analysis of wastage and is worth
mentioning because of the deviation of this model from the conventional
models relating to the analysis of wastage in manpower system. The
author of this paper has presented a model which is designed to
investigate the effect on wastage of the internal structure of the company
and the promotion experience of its staff. Also a stochastic model has
been developed to depict the probability of the number of promotions to
an employee in the interval (0, t]. The estimation method for calculating
the probabilities is also discussed.
Mukherjee and Chattopadhayay [58] have discussed an optimal
recruitment policy. The authors have considered an organization in which
number of persons are recruited at time t . Every recruited person can be
in service for ' t ' years at the most. It is also assumed that the efficiency
of each recruited person is adversely affected by the longer duration of
service. The authors have derived a recruitment policy at interval of time
t . The optimal values of ' t ' which minimizes the total cost of unified
vacancies and forced recruitment have been worked out.
Gardner [28] has presented a research study on exponential
smoothing where its historical development was traced to the time of
second World War. The research critically commented on the merits of
various models and deferred with others based on his research as well as
39
research of other researchers. In the conclusion exponential smoothing
technique was also rated as one of the best forecasting methods.
An optimal planning of manpower training programmers has been
analyzed by Goh et. al [29]. Two different models are discussed in this
paper by assuming the finite planning period and an infinite planning
period. A finite slate Markov chain is used to model the manpower state
for the finite planning period and the optimum solution is computed
using the dynamic programming technique. A non-linear integer
programming problem is used to model the manpower state for a finite
planning period.
A model responding to promotion blockage is discussed by
Kalamatianou [42]. This model is proposed for manpower system in
which promotion probabilities are functions of seniority within grades.
Poornachandra Rao [63] has made attempts to identify various
costs associated with manpower planning system. Based on this a
manpower planning model with the objective of minimizing the
manpower system costs was formulated. The major limitation of the
model is the consideration of manpower costs in isolation of various
constraints and operating policies under which a manpower system
operates. The model proposed an integrated model which will minimize
40
the manpower costs in the presence of the system constraints and other
operating policies.
Raghavendra [64] has discussed a bivariate for manpower planning
system. The author indicates that in many developing countries there is
limited mobility of people from one organization to another. This results
in policies of promotion and recruitment, which will have long term
effects on the organization. It is also observed that in many organizations
especially in the public sector undertaking two types of policies on
promotions are followed (1 ) promotion by seniority ( length of service ),
(ii) promotion by performance rating. The author has taken the two
aspects in a bivariate frame work. So a Markov chain model is developed
to derive the estimates of the transition probabilities. It is also shown that
in presence of organizational objectives the promotional policies can be
translated in to respective levels in terms of either seniority or
performance rating or a combination of both. The author has obtained the
joint probability distributions of the two random variables namely x is the
seniority and y is the performance rating; the marginal distributions are
also obtained. The minimum level of seniority required for promotion
and the minimum level of performance rating required for promotion
have been estimated. Numerical examples are also furnished to explain
the usefulness of the model.
41
According to Barthlomew et al, [11] the analysis of individual
difference is of fundamental importance in the study of manpower
system, in particular, wastage (loss from the system). Any attempt to
describe wastage pattern must reckon with the fact an individual's
propensity to leave a job depends on many factors, both personal and
environmental.
McClean [52] has discussed a Mathematical model using which
provides a method to predict the growth of manpower needs of the
Northern Ireland Software Industry. Northern Ireland has been in an
enviable position regarding the quality and quantity of manpower in its
software industry. The growth of the software industry in turn results in
increased demand for manpower in IT industries. So the author has taken
two groups of staff (i) those who are working in IT firms (ii) those who
are working in firms which are IT users. Growth in the demand for
software personnel in IT firms depends upon the developments achieved
in software industry. The growth of manpower in IT firms will be
primarily dependent on factors specific to software sectors. The author
has used a transition model based on Markov analysis and using the
model the author has prepared a formula for predicting the demand for
manpower in the IT industry. This model will be very useful to take some
steps well in advance so that the training and recruitment could be
42
suitably modified and planned. The author has given the projection
figures for a period of 5 years in the future and it would give an idea of
the need for training sufficient number of personnel well in advance.
Subramaniam [77] has studied the optimal time for the withdrawal
of the voluntary retirement scheme in manpower planning. A period of
length T years is considered during which the employees are permitted to
go on voluntary retirement on k selected epochs. As and when the staff
strength reaches a level, which is called threshold level the voluntary
retirement scheme is withdrawn because the staff strength reaches the
required level. The optimal strength of T years which is numerically
illustrated with graphs is obtained based on certain assumptions.
An analytical model which deals with finding the size of each grade
in a hierarchical organization has been developed by Kenway et. al [43].
The size of each grade and proportion of employees in each grade are
derived using a number of assumptions regarding the demand in that
organization, the wastage, rate of retirement, etc. The concept of
promotion is also taken into account. It is assumed that the demand for
manpower increases exponentially at a rate 'p' The wastage rate ,w x t
for employees of age x at a time ' t ' is defined as the proportion ,w x t
t of staff at age x at time t that leaves the organization tt t N N t
43
is the total size of the system at time ,t p t is taken to be the proportion
of the employees in the top grade. R t 5t denotes the total number of
recruits in the interval tt t . A constraint taken up is
t
op t xt g xt dt . Where ,f x t is defined as the population of all
employees age x at time ' 't who have been promoted to the top grade
and ,g x t is the age distribution of all employees at time t . The
constraint that demotion is not permitted is represented as , f x a t a
for all ,x t and a . Using all the above the authors have obtained a model
for the sizes of the different grades at future point of time ' '.t
Thio [80] has discussed the need for retention strategies especially
in turbulent times. According to the author it is commonly assumed that
the retention of staff in an organization is an indicator of the health of the
organization, as well as its unhealthy state. But it is difficult to establish
the links between attrition and unhealthy state. The author has made an
explanatory attempt of synthesizing some ideas about attrition and
retention. To some extent the staff turnover is inevitable and cannot be
beneficial. The process of attrition makes way for the recruitment of new
blood and also facilitates the career progression of those who remain in
the organization. However high and unexpected turnover can be a
reflection of negative job attitudes and low staff morale. It may warrant
44
counter measures. Remedial measures are necessary to manage attrition
in a way that causes least dislocation to the work of the organization.
The author has also indicated that the reduced staff strength due to
attrition will result in a loss of customers or clients to the organization.
The author has taken up the issue of attrition with particular reference to
certain welfare organizations in Singapore. The author has reiterated
certain retention strategies such as recognition of work turnover by the
staff as a key measure which helps in the management of attrition. The
management must provide participative opportunities available for the
staff. It must develop a congenial environment to work for the staff.
There must be informal communication among the staff, job rotation and
recognition for work turnover, etc. Retention of key staff requires
continuing leadership influence and management affiliation. Even though
attrition is not always harmful, it should not be dealt passively.
Estimation for semi-Markov manpower models in a stochastic
environment has been discussed by McClean and Montgomery' [53].
Manpower planning, often needs to estimate and predict distributions of
duration in various grades in the hierarchical set up of a company. A
methodology for fitting a stochastic environment to manpower data for
both the non-homogeneous semi-Markov system in a stochastic
environment (S-NHMS) and the non-homogeneous semi-Markov system
45
in a stochastic environment (S-NHSMS) is discussed in this paper. These
models provide means of describing changes in the environment between
contiguous periods, in particular homogeneous time period and the
process governing movements between such periods. Thus providing a
description of time heterogeneity. The predictions of future movements
of staff through the system are made.
Koley [44] has brought out the importance of human resource
investments in order to place any organization in a comfortable position
and on the appreciating track. This paper suggests that to build up the
human resource, investments on employee recruitments, training and
development besides cost arising due to wastages and salaries may be
used for decision making purpose. It suggests the use of available tools
and techniques like, works study, learning curve, activity based costing,
decision tree and risk analysis, life cycle cost approach to assess the cost
of making managers as investments. A company where there is no dearth
of qualified manpower can be one among the richest in the world to build
up the manpower.
In many organizations, the number of sanctioned positions may be
vacant year after year. Huge amount is spent by many organizations for
the requirement of specialists as well as training and development of such
persons. The recruitments process involves locating the right type of
46
candidates from inside and outside the organization through interval
circulars, external advertisements etc.
The author suggests that the expenditure on the projects on HR
development and related activities should be carefully decided so that the
control of cost over HR can be decided by using PERT and CPM
methods. It is also necessary to measure the HR productivity.
It becomes vital to decide the effectiveness of various strategic
moves of human resource managers from time to time. The build up of
HR costs and investment figures is not to put the man on the balance
sheet but to use those for decision making purposes.
Gupta and Kundu [32] have studied some properties of a new
family of distributions, namely Exponentiated Exponential distribution.
The Exponentiated Exponential family has two parameters (sales and
shape) similar to a Weibull or a Gamma family. It is observed that many
properties of this family are quite similar to those of a Weibull or a
Gamma family; therefore this distribution can be used as a possible
alternative to a Weibull or a Gamma distribution. Some numerical
experiments are performed to see how the maximum likelihood
estimators and their asymptotic results work for finite sample sizes.
47
Sathyamoorthy et al [70] have discussed a manpower model for
estimating the propensity to leave the primary job. In this paper they have
discussed the method of deriving the propensity of individuals to leave
the primary job in an organization which leads to attrition. Cox's
regression approach has been used to derive the level of propensity of an
individual in a primary job in an organization. The authors have taken up
the exponential distribution as a special case to estimate the propensity to
leave the organization. The specialists holding the prmary job have some
covariates of personnel character. These covariates also contribute to the
intensity or degree of propensity to leave the job. In addition to the
degree of propensity generated by the Completed Length of Service
(CLS) in the organization, the covariates also contribute and hence the
combined influence of both namely CLS and the covariates decide the
degree of propensity to leave the job.
Sathiyamoorthy and Parthsarathy [68] have considered a two grade
organization in which the mobility of personnel from one grade to the
other is permitted as to compensate the loss of manpower which is larger
among the two grades. They have considered the case in which the Max
2|Y Y is taken to be the threshold level of the organization where iY
andY2 are the individual thresholds of the grades respectively. They have
48
obtained an expression for the expected time for recruitment in a two
grade organization.
Sathiyamoorthy and Parthasarathy [69] have used the idea of
change of prarmeter for the threshold distribution after the truncation
point. This idea is similar to SCBZ property where the parameter
undergoes a change. Assuming the truncation point is itself a random
variable following exponential distribution, which is taken for the
threshold level. The expected time for recruitment is also obtained using
the shock model approach and the results are compared when there is no
change of parameters for the threshold distribution.
Charles et al., [16] have examined the interaction effects of
maintenance policies on batch plant scheduling in a semiconductor water
fabrication facility. The purpose of the work is the improvement of the
quality of maintenance department activities by the implementation of
optimized preventive maintenance strategies and comes within the scope
of total productivity maintenance strategy.
The production of semiconductor devices is carried out in a water
lab. In this production environment equipment breakdown or procedure
drifting usually induces unscheduled production interruptions.
Jeeva et al., [41] have discussed frequent wastage or exit of
personnel common in many administrative and production oriented
49
organizations. Once the accumulated number of exits from the
organization reaches a certain threshold level, it could be viewed as a
"breakdown point". The time to attain breakdown point is an important
characteristic for the management of the organization. A shock model
approach is proposed to obtain the expectation and variance of the time to
attain the threshold level.
Elangovan et al., [21] have discussed a model using which the
optimal level of hiring expertise service in manpower planning has been
discussed. It has been assumed that the cost of hiring experts in some
chosen areas of human activity is fairly high and hence it would be
advantageous to use the Mathematical methods to find the optimal
duration for which the hiring of experts in terms of man hours. If the
number of hours of contract is more than the requirements it would be a
financial loss. Again if the number of hours of contract is below the
requirements or demand it would prove to be financial loss due to
shortages. Hence taking the demand into consideration the exact or
optimal size of number of hours of purchase is determined. In doing so
the demand for man hours of expert service is assumed to be a random
variable and is following the so called exponential distribution.
Appropriate costs of excess manpower as well shortage costs are
assumed in obtaining the optimal solution. Another interesting variation
50
in this model is also discussed in this paper. It is further assumed that the
demand for expert manpower undergoes change from time to time. Also
if the demand for expert manpower is beyond a particular level then the
cost of hiring also undergoes a change. All these modifications and
additional assumptions make this model very much in agreement with
real life situations. For each kind of the situation that arises in practice
the optimal policy for hiring expertise has been obtained by the authors.
Numerical examples of different types are taken up and the situation of
optimal types are derived and the graphs are provided.
Sureshkumar [78] has developed a stochastic model in which the
prediction of the likely time to recruitment due to the depletion of
manpower in a two grade organization. The manpower planning studies
about depletion of manpower due to leaving of personnel, known as
attrition. This is also called as 'wastage' The attrition takes place on
successive occasions of policy decisions regarding pay revision,
perquisites and when targets are fixed. The recruitment is not taken up on
every occasion of attrition, but the deficiency in manpower is managed
by transfer of persons from one grade to the other where the attrition is
more pronounced. The authors have also introduced the concept of
threshold level for cumulative attrition. If the attrition or wastage crosses
the threshold level then the recruitment has to be done. This is with a
51
view to reduce the cost of frequent recruitments. The various costs
involved in recruitment are discussed in detail by Poornachandra Rao
[63]. The expected time to recruitment is predicted assuming the wastage
as random variable on successive occasions of attrition. The authors have
discussed two such models. In the first model the transfer of personnel
from one grade to another is permitted. In the second model the transfer
is not permitted. These mathematical models serve as projection
techniques so that the management can adopt suitable strategies to
contain the level of attrition and also decide suitable policies to deal with
the consequences. Numerical illustrations are also provided to support the
findings.
Anantharaj et. al [4] have discussed on the method of arriving at
the optimal time intervals between recruitments. When attrition takes
place on successive occasions over a period of time and cumulative
attrition when reaches a particular level called the threshold the
recruitment becomes necessary to make up the loss of manpower.
Recruitments very often are not advisable since it involves costs of
several nature. Hence the determination of the optimal time interval
between recruitments, become necessary. In deriving the optimal time
periods between the recruitments, the authors have used the shock model
and cumulative damage process approach. The cost arising due to the
settlement of gratuity and other compensation packages, the cost arising
52
due to breakdown of regular work schedule are all taken care of in
obtaining the optimal solution for the problem. Numerical illustrations
have been provided on the basis of simulated data to prove the validity of
application of this model and also the behaviour of the optimal solution
obtained when changes take place in the influencing variables.
Laura Roe [47] has indicated that a twenty five percent IT industry
average turnover rate persists all over. This requires the recruitment and
hiring of same number of employees to make up the gap. Some
suggestions as how to improve the retention rates are suggested by the
author. Many of them may look impossible, but are critically important
from the view of retention strategies. Some suitable strategies suggested
for promoting retention are:
The hiring process should be as good as possible, since retention
starts with good hiring process.
The technical staff should be motivated and told how project is, to
their team and company.
Training is important factor that contributes to the retention.
Technicians especially those at the risk of leaving should be
assigned technically challenging tasks.
53
If necessary internal assignments should be assigned to use leading
edge technologies.
Managers and IT staffer relationship have a profound effect on
retention. Provision of high quality work place and corporate atmosphere.
In addition to the above the author has suggested a number of other
retention strategies.
Srinivasan and Sudha [76] have considered four grades
organizations with policy of recruitment. The mean and variance of time
to recruitment are derived by assuming random threshold following non
identical exponential distribution for each grade and the threshold for the
organization as the minimum of four thresholds.
Mallikarjunan [51] has given an overview of the causes and
remedies for employee attrition. According to the author employee
attrition is caused not only by natural inevitabilities like disability, death,
retirement and resignation, but also by the burgeoning mobility of human
resources or the human capital. One of the toughest problems that
confront HR managers is employee attrition. Due to the vertical growth of
the business. Process, services and products, skilled and even semi skilled
workers find a matrix of possible avenues for self development.
54
The author has indicated that the nature of the business, the nature
of responsibility shouldered by them is the reasons for the attrition of
employees, This is very much relevant to the software industry. He also
indicated that the employee attrition can be classified into two categories
namely (i) drive attrition, which is caused by the policy practice and
treatment of the employer in the industry,(ii) Drag attrition as a result of a
number of uncertainties faced by the employee in the working
environment such as absence of opportunity for advancement in career,
absence of opportunity to achieve mental and functional growth.
A few industries have been found to be in the constant threat of this
syndromes of attrition and they are Information Technology and
Hardware Industry. Information Technology and Software Development
Industry Call Centres. Business process outsourcing industry. Other
industries like pharmaceuticals, manufacturing, etc.
Arivazhagan et al [5] have developed a mathematical model which
can be used to estimate the likely time at which the enrolment for
recruitment should be stopped. According to the authors these are many
organizations which are providing service in the HRM sector. The supply
of skilled laborers and specialists is one of the main areas of activities in
such organizations. They keep a reserve or inventory of skilled personnel
and whenever there is demand or request from organizations or industries
55
the supply is the main activity. The enrolment has to be stopped at a
particular level called optimal enrolment. Assuming that the allowable
level of enrolment as the threshold level the expected time to stop
enrolment has been found out. Numerical illustrations are also given.
Suvro Ray Chaudhuri [79] has given a detailed account of
employee attrition and the methods of predicting the attrition rate and
also the strategy for mitigation of the rate of attrition. According to the
author the manpower attrition is similar to the customer switching
problems in the case of products. The author has used the Markov
analysis to predict attrition. Human resource professionals are under
increased pressure from a different kind of a corporate problem which
causes no less harm to human capital assets. The American Productivity
and Quality Centre (APQC), has made three different categories of
knowledge that suffer due to attrition. They are as follows:
Cultural knowledge. This includes management practices, values,
respect for hierarchy, and decision flows.
Historical knowledge - This includes the organization's journey
from the day it was founded till the present.
Functional knowledge - This includes technical, operational.
Process and client information.
56
From the organization point of view the counter strategy is to
predict attrition zones which depend on the critically or type of
knowledge, that is important to organization and there by evolve plans to
counter loss of human assets from those positions. The attrition is one of
the main areas in the field of knowledge management because it is easier
to fill up any position by recruitment but filling the knowledge gap is not
easy. The author says that the organizations have to spend huge sums of
money on recruitment. This is due to the fact that the functional
knowledge of the new persons may not be equal to that of the person who
has left the organization. The author has used the Markov Analysis by
taking the employees as internal customers. The purpose of Markov
Analysis is to predict the rate of attrition based on the present data. The
transition probability matrix which is a basic tool in Markov Analysis is
also discussed and the solution is derived to predict the future rate of
attrition from the organization.
It is quite reasonable to think that recruitment cannot be done as
and when manpower leaves, a threshold can be kept upto which loss of
manpower can be allowed, after then the recruitment can be done. This
idea of recruitment to start after the depletion of manpower reaching a
threshold is brought in the mathematical model of R. Elangovan, R.
Sathyamoorthy and E. Susiganeshkumar[22]. The expected time to
57
recruitment is derived in this model As the exit of personnels is
unpredictable, J. B. Esther Clara and A. S. Srinivasan [24] have
constructed a mathematical model of new bivariate recruitment policy
involving optional and mandatory thresholds for the loss of manpower in
the organization, based on shock model approach and cumulative damage
process to enable the organization to plan its decision on recruitment.
Assuming different distributions for optional and mandatory thresholds,
expected time to recruitment is obtained.
Based on shock model approach, a mathematical model is
constructed by J.B.Esther Clara and A.Srinivasan [25] using an
appropriate univariate max policy of recruitment and an analytical
expression for the mean time to recruitment is obtained under suitable
conditions on the loss of manhours, inter-decision times and thresholds.
Ishwarya. G., Mariappanan. P and Srinivasan .A. [37] in their
paper have discussed the problem of time to recruit when the thresholds
follow extented exponential distibution with shape parameter 2 which is
more general than exponential distribution.
M. Jeeva and Fernandes Jayashree Felix [41] have developed a
manpower model where the recruitment process follows a pre-emptive
repeat priority service discipline. The transient behaviour of the
applicants waiting for the recruitment process is discussed in this model
58
and mean and variance of the high priority and low priority applicants are
determined.
When the Manpower System of an organization is exposed to
Cumulative Shortage Process due to attritions that cause manpower loss,
breakdown occurs at threshold level. S.Mythili and R.Ramanarayanan
[60], in this paper have considered the Manpower System with two
groups A and B. Group A consists of man-power other than top
management level executives. Group B consists of top management level
executives. Group A is exposed to cumulative shortage process and
shortage process of group B has varying shortage rates. Recruitment is
done to all the shortages of the two groups. The expected lime to recruit
and recruitment time are determined.
S.Mythili and R.Ramanarayanan [59], have considered manpower
system of an organization for which Attrition Reduction Strategy (ARS)
is applied prior to recruitment. In this paper recruitment policy of filling
vacancies one by one and parallel recruitment policy of filling vacancies
simultaneously are considered.
Nabendu Sen and Manish Nandi [61] have formulated a strategic
planning using the Goal Programming approach to Rubber Plantation
Planning in Tripura
59
A project of a company requires processing in several stages for
completion of the same. K.Usha, A.C.Tamil Selvi and R.Ramanarayanan
[81] have considered n intermittent stages and the project visits and
revisits these n stages before it moves to the completion stage 1n . The
probability generating function of the number of paths of specific type
namely x to y the project executes before completion is determined. Its
expected value and the variance are found. Here every path x to y is
treated as a recruitment of a batch of employees. K. Usha,
P.Leelathilagam and R.Ramanarayanan [82] have Considered a project of
a company with n intermittent stages. The project visits and revisits these
stages before it enters the completion stage 1n . Five stages have been
considered where one stage indicates changes in company policy for
manpower, one stage indicates project modification for manpower, one
stage indicates shortage of manpower, one stage indicates recruitment of
manpower and one stage indicates training of manpower. In this paper, is
obtained the probability generating function of the number of pentagonal
loops the project forms in the respective stages before completion. Its
expected value and the variance are determined.
Vinod Kumar Mishra and Lai Sahab Singh [88], in this paper, a
deterministic inventory model is developed for deteriorating items in
which shortages are allowed and partially backlogged. Deterioration rate
60
is constant. Demand rate is linear function of time, backlogging rate is
variable and is dependent on the length of the next replenishment. The
model is solved analytically by minimizing the total inventory cost.
61
CHAPTER 3
MANPOWER PLANNING PROCESS WITH TWO GROUPS
USING STATISTICAL TECHNIQUE
3.1 INTRODUCTION
Employees are the most important asset for a business. They serve
to create or promote an organization's culture, and they significantly
affect the success of a business. In challenging economic times, the cost
of hiring inefficient personnel may prove to be detrimental to the
profitability of an organization. An effective and thorough manpower-
recruiting process requires an employer to carefully choose the most
talented employees who will positively benefit the organization or
business. A needs analysis initiates the manpower recruiting process. This
phase entails identifying a vacant position or creating one to meet new
needs that have arisen in the organization. this may be an entry mid- or
upper-level management position. The employer then develops a job
description describing the duties involved with this position. Criteria such
as skills and competencies, experience, age, and education that best serve
the position are also identified. Using this information, the employer
prepares a standard application form to collate information provided by
the applicants, in addition to their own resumes. The vacancy is then
advertised.
62
3.2 DESCRIPTION AND ANALYSIS OF THE MODEL
In this sectiona two graded manpower system is considered and the
description of the model is given below.
Assumptions
1. Group A is given at the most k observation times each with
exponential distribution with parameter ' ' before recruitment. On
completion of the first exponential observation time, recruitment is
done with probability , or, the second observation starts with
probability , where 1. The process is repeated upto i
observations for 1 1.i k On completion of the thk observation,
recruitment is done with probability 1 . If 1T is the time to recruit due
to group A and 1 2; , ..X X are the shortages caused by manpower
loss in group A, then, 1
1
k
j
j
T X
with probability 1k for 1 1i k
or, 1
1
k
j
j
T X
with probability 1k Group B has shortage process with
varying shortage rates. At time 0, the shortage rate of the group is .
Let 2 be the time at which breakdown of group B occurs
necessitating immediate recruitment.
2. Recruitment for MPP starts if either of the groups A or B has a
breakdown. All the shortages due to manpower loss are compensated
by recruitment.
63
3. When recruitment is done due to breakdown of group A, recruitment
time corresponding to the thi observation is ,iR 1 i k When the
breakdown occurs due to group B, recruitment is done for shortages in
group B and also for shortages in group A for the number of
observations completed. All the recruitment times are independent and
identically and distributed random variables with distribution function
R y such that
( )
y
o
ydR y .
3.3 MAIN RESULT
Based on the assumptions, recruitment starts at time
1 2min ,T T T . Identifying the exponential phase time of the
modified Erlangian, the pdf of time 1T is given by
12
1
0
( )! 1 !
i kk
x k x
i
i
x xf x e e
i k
(3.1)
2
,P T x Rt yx y
2( )1 ( ) ( ) ( )
1!
x xxH x e r y e r y
22 3( )
( ) ....2!
xxe r y
2 12 ( 1) 1( ) ( )
( ) ( )( 2)! ( 1)!
k kx k k x k kx x
e r y e r yk k
(3.2)
64
The first term corresponds to breakdown due to group A and the
second terms corresponds to breakdown due to group B Using (3.1) and
(3.2), we get,
2
,P T x Rt yx y
1 2
21*
0 !
ik
ix x x i
i
xe e e r y
i
1 2
1
1 *
1 !
k
x x x k kxe e e r y
k
1 2
1* 1
1 2
0 !
ik
ix x x i
i
xe e e r y
i
(3.3)
(3.3) can be simplified by taking Double Laplace transform.
T RtE e e
1
1 1
1 1 1
( ) ( ) ( )
( )
k k
r r r
r
1
2 2
2 2 2
( ) ( ) ( )
( )
k k
r r r
r
1 1
1 2
( ) ( ).
k k
k kr r
(3.4)
for 0 and 0 , we obtain from (3.6)
( )tE e
1 1
1 2
1 2
1 1
k k
65
1 1
1 2
k k
k k
1 2
1 2
1 2
1 1
.
k k
(3.5)
RtE e
1
1 1
1 1 1
( ) ( ) ( )
( )
k k
r r r
r
1
2 2
2 2 2
( ) ( ) ( )
( )
k k
r r r
r
1 1
1 2
( ) ( )k k
k kr r
(3.6)
From (5) and (6), by differentiating,
1 1 2 2
1 1( ) 1 1 .
k k
E T
(3.7)
1 1
1 1 2 2
( ) ( ) 1 1 1
k k
E Rt E R
(3.8)
3.4 NUMERICAL ILLUSTRATION
By giving different values to the parameters in E T and E Rt
by varying λ from 1 to 10, we present the graphs of E T and E Rt .
66
Table 3.1
1 0.07, 2 0.05, 0.4, 0.6, 2,k ( ) 3E R
1 2 3 4 5 6 7 8 9 10
E T 3.479 1.923 1.320 1.003 0.809 0.678 0.583 0.512 0.456 0.411
E Rt 2.71 3.32 3.45 3.35 3.29 3.25 3.22 3.19 3.17 3.16
67
Table 3.2
1 0.07, 2 0.05, 0.6, 0.4, 2,k ( ) 3E R
1 2 3 4 5 6 7 8 9 10
E T 2.876 1.566 1.069 0.810 0.652 0.546 0.469 0.411 0.366 0.330
E Rt 2.640 2.500 2.431 2.385 2.350 2.321 2.298 2.278 2.260 2.245
3.5 CONCLUSION
From tables 3.1 and 3.2, we observe the behavior of E T and E Rt
i.e., mean time to recruit and mean Recruitment time for fixed values of
1 2, , , ,k and E R . When the parameter λ increases, the value of
E T increases and E Rt decreases. When α increases, both E T and
E Rt decrease.
68
CHAPTER 4
OPTIMAL MANPOWER RECRUITMENT AND PROMOTION
POLICIES FOR THE TWO GRADE SYSTEM USING
DYNAMIC PROGRAMMING APPROACH
4.1 INTRODUCTION
Bartholomew and Forbes (1979) have described the state of the art
I n various facts of manpower planning. Edwards (1983) has surveyed
various models on their assumption and application and concluded that
good presentations of results are more important than theoretical
sophistications. Price and Piskor (1972) have developed Goal
programming model of manpower planning system for financial,
manning, promotion and manpower accounting.
Zanakis and Maret (1981) have formulated a Markovian goal
programming model with pre-emptive priorities and provided a more
flexible and realistic tool for manpower planning problems. Mehlmann
(1980) has developed optimal recruitment and transition strategies for
manpower systems using dynamic programming. He has formulated a
dynamic programming recursion with the objective of minimizing a
quadratic penalty function which reflects the importance of correct
manning of each grade under preferred recruitment and transition
patterns.
69
While the models developed in the manpower planning literature
have considered financial and labor costs and the system and resulted in
the form of recursive optimization. A dynamic programming model has
been found to be analogous to the Wagner–Whitin (1958) model, based
on the cost data, it generates the optimal recruitment and promotion
schedules for future periods.
4.2 MANPOWER SYSTEM COSTS
Manpower system costs depend upon the various factors. The
various costs associated with manpower system consist of the following:
i) Recruitment and Promotion costs
ii) Overstaffing costs Understaffing costs Retention costs Wastage
costs.
iii) Recruitment and promotions costs
iv) Cost of advertising
v) Cost of conducting written test
vi) Cost of information processing
vii) Cost of manpower working on the processing of application
viii) Cost of administrative authority which determines recruitment
and promotion policies
ix) Costs incurred in the form of payment to the interview
committee members or the wages of the people on the interview
committee.
70
x) T.A paid to the candidates which is optional.
xi) Cost of medical examination done by the organization
xii) Cost of training people
xiii) Miscellaneous expenditure, including postage, telephone calls etc.
The actual components of recruitment and promotion cost depends
upon the procedure followed by the organization for recruitment while
the above components are indicative only. Even though the charges are
paid by applicants for processing, it is not proportional to the
actual recruitment and promotion costs met by many organizations.
The cost of advertising and cost of administrative authority from a
fixed component is independent of people recruited or promoted based
on the suitability of the candidates.
The costs like traveling expenses are paid to interviewing people
and also depend on the policy of each organization in determining the
number of candidates to be interviewed. According to management’s
policy, if the people to be called are a predetermined ratio, which is
proportional to number of candidates selected or interviewed and
remains constant.
A fixed and a variable component per recruiter or promoter is
applicable for all the costs like conducting written test, manpower
working on the processing of applications, Information processing,
71
medical examination and training the people. The fixed costs are higher
if the selection process is in groups like military recruitment process.
(a) Overstaffing Costs
Overstaffing costs are those incurred owing to an unutilized
workforce. These costs are analogous to the inventory costs in a
production / inventory situation.
(b) Understaffing Costs
Understaffing costs are those resulting from decreased
productivity and loss of goodwill (in a profit-motive organization) as a
result of the non-availability of the workforce.
(c) Wastage Costs
The costs result from the retrenchment or retirement of the
employee.
(d) Retention Costs
There are certain costs which are involved in retaining an
employee in an organization. These costs consist of (i) probation costs,
(ii) training and development costs, and (iii) internal mobility costs.
Probation costs are those incurred owing to the learning effect of
an employee during a probationary period. The training and
development costs are different from the recruitment costs and are
72
incurred owing to the development programmes which an employee
undergoes during the course of his service to the organization. Internal
mobility costs are the costs involved in demotion or transfer of an
employee within the organization.
4.3 MATHEMATICAL MODEL
The following assumptions are made while formulating the
manpower planning problem to determine optimal recruitment and
promotion policies:
The recruitment and promotion size are known and fixed.
Recruitment and promotion at a particular grade is considered.
Recruitment, promotion and overstaffing costs are known and fixed.
Understaffing is not allowed in both the grades.
Notations
1. :R t Recruitment in any period .t
:S t Fixed recruitment cost in period .t
:P t Promotion at any period .t
:Q t Cost of promotion / period.
:i t Cost of overstaffing per recruiter or promoter per period.
:l t Number of people recruited / promoted in an earlier
period for the requirements of period .t
1 :x t Number of people recruited in period t at Grade 1.
2 :x t Number of people recruited in period t at Grade.
73
2. :y t Number of people promoted in period t from Grade 1 to
Grade 2.
1 :v Variable cost of recruitment at Grade 1 / employee
recruited.
2 :v Recruitment at Grade 2 / employee recruited.
:u Variable cost of promotion at Grade 1 to Grade 2.
Overstaffing cost not allowed for Grade 2 since it was for higher
level, not necessary. Since, we need to satisfy all requirements on time,
so that understaffing is prohibited.
The requirement cost in period t is given by the conncave function:
1 , 1,2.i iS t x t v x t i (4.1)
The promotional cost in period t is given by:
Q t y t uy t (4.2)
The overstaffing cost is i t I t . The total cost of recruitment
for the T — period planning interval is:
1
T
i i i
i
S t x t v x t i t I t
(4.3)
The total cost of promotion for the T — period planning interval is:
1
T
t
Q t y t uy t i t I t
(4.4)
Thus the total cost of recruitment and promotion for the T — period
planning interval is:
74
1
, 1,2.T
i i i
i
S t x t Q t y t v x t uy t i
(4.5)
Here we take 0 0 0i l without loss of generality.
The problem is to minimize this sum, subject to the constraint that
all recruitments and promotions must be met on time, and since the
variable cost of recruitment and promotions are constant. we have that
are constant in (3)
1
T
i
t
Vv R t
and 1
T
t
u P t
are constants
Thus the problem may therefore be stated as Minimize
1
, 1,2.T
i
i
S t x t Q t y t i t I t i
(4.6)
Subject to
1 1
, 1,2,... : 1,2t t
ik k
k k
x R t T i
and
1 1
, 1,2,...t t
k k
k k
y P t T
4.4 DYNAMIC PROGRAMMING FORMULATION
Theorem
The well-known Wagner—Whit in model is characterized to
determine economic lot size with this model. The fixed recruitment
and promotion cost is analogous to the set-up cost and the overstaffing
75
cost is analogous to the cost of carrying inventory in an inventory
system.
The propositions of Wagner — Whitin model, which facilitate
formulation of Dynamic programming recursion are thus given.
Theorem 1
There exist an optimal program such that:
0iI t x t for all t and 1,2.i
Theorem 2
The minimum cost policy has the property that the recruitment
cost x takes the values 0, , 1 ,...,( ) ( ) ( ) ( ) ( ) ( ) 1 ... R t R t R t R t R t R T
and the promotion cost y takes the values ( ) ( )0, , 1 ,...,( )P t P t P t
( ) ( )1 ... ( ).P t P t P T
Table 4.1
Hypothetical Data for a 5 Year Planning Period of a Manpower System
Year R P S in 000’s Q in 000’s I in 000’s
1 79 41 728 540 15
2 34 10 705 220 12
3 52 14 698 385 16
4 61 38 714 412 14
5 25 8 708 398 16
76
Theorem 3
There exist an optimal program such that if R is satisfied by
some **x t and P is satisfied by some ** , ** *,y t t t then R t
and , ** 1,..( ) / ., * 1P t t t t are also satisfied by **x t and **y t .
Theorem 4
Given that 0I t for period t , it is optimal to consider periods
1 to 1t by themselves. Let F t denote the minimal cost program
for periods 1 to t, then:
F t 1
11
min 1 .,t t
j tn j k h
S j Q j i h R k P k F j
1S t Q t F t (4.7)
The above recursion, stated in words, means that the minimum
cost for the first ‘t’ periods comprised a fixed recruitment and promotion
cost in period ,j plus the charges for satisfying requirements R k and
promotion , 1,...,( )P k k j t by recruiting and promoting manpower in
period ,j which results in overstaffing cost, plus the cost of adopting an
optimal policy in periods 1 to 1j taken by themselves. We state
below the manpower planning horizon theorem analogous to the
Wagner — Whitin planning horizon theorem which further amplifies
determination of optimal policies.
77
t
4.5 THE MANPOWER PLANNING HORIZON THEOREM
If the minimum in (1) occurs for ** *j t t at any period t, then
in periods *t t it is sufficient to consider only ** .t j t If * **t t
then it is sufficient to consider programmes such that ( )* 0x t
and ( )* 0y t . Wagner—Whitin algorithm can be made use of to
determine the optimal recruitment and promotion policies. The
algorithm at period *, * 1,2,...t t N may be stated as:
Consider the policies of recruiting and promoting at (period
**, ** 1,2,.. . *.t t t
The total cost of these t* different policies by adding the
fixed recruitment cost, promotion cost and overstaffing costs
associated with the recruitment and promotion at period t** and the
cost of acting optimally for periods 1 to ** 1 t considered by
themselves. The latter cost has been determined previously in the
computations for periods 1,2, * 1.t t
(3) From the t* alternatives, select the minimum cost policy for
periods 1 to t* considered independently.
(4) Proceed the process to period * 1t or stop if * .t N
78
Table 4.2
The Calculations of the Manpower Planning Problem Presented
Year 1 2 3 4 5
S 728 705 698 714 708
Q 540 220 385 412 398
i 15 12 16 14 16
R 79 34 52 61 25
P 41 10 14 38 8
1268 1928* 3011 3846* 4952
2193 2720* 5690 4308*
4595
Minimum cost 1268 1928 2720 3846 4308
Optimum policy 1 2 2.3 4 5
4.6 NUMERICAL ILLUSTRATION
Table 4.1 shows the hypothetical data for a 5 year planning period
of a manpower system. Table 4.2 summarizes the calculations of the
manpower planning presented in Table 4.1. Thus the optimal policy may
be stated as follows: Recruit and promote in period 4,
4 4 86 46 132x y and use the optimal policy for periods 1 to 4,
implying (2) Recruit and promote in period 2, 2 2 86 24 100x y and
use the optimal policy for periods 1 to 2, implying (3) Recruit and
promote in period 1, 1 1 79 41 120.x y The total cost of this policy is
4308.
79
4.7 CONCLUSIONS
In this paper an attempt has been made to obtain the optimal
number of recruits and promotions made so that the total cost incurred is
minimum in the manpower planning system along with the various costs
like recruitment costs, promotion costs, overstaffing costs, wastage costs
and retention costs. There are two types of cost have been taken into
account namely fixed and variable costs. The model has been found to
be analogous to the Wagner-Whitin model in a production or inventory
situation. The major limitation of the model is the fact that it is
considered in isolation from the various constraints and operating
policies under which a manpower system operates. As another constraint
of the model is that, it is assumed that there is no overstaffing in the
higher grade. This model can also be discussed without this constraint as
further work.
80
CHAPTER 5
EXPECTED TIME FOR RECRUITMENT IN A TWO GRADED
MANPOWER SYSTEM ASSOCIATED WITH CORRELATED
INTER-DECISION TIMES – A SHOCK MODEL APPROACH
5.1 INTRODUCTION
For a single graded manpower system, Sathiyamoorthy and
Elangovan (1998) have obtained the mean and variance of the time to
recruitment assuming that the random variables
denoting staff depletion are independent and identically
distributed random variables, using cumulative damage model
concept without using any cost structure. Mariappan and
Srinivasan (2001(a)) have extended the result. when the
interdecision times are exchangeable and constantly correlated
exponential random variables for a single graded system. For a
two grade system, Sathiyamoorthy and Parthasarathy (2002)
have obtained the mean time to recruitment under suitable
conditions.
In this chapter, an organization with two grades each
having its own threshold level is considered. Two mathematical models
are considered employing two different univariate policies of recruitment.
In model 1, recruitment is made whenever the threshold crossing
takes place in both the grades and in model 2, recruitment is made
81
whenever the threshold crossing takes place in any one of the grades. The
objective of this chapter is to find the mean and variance of time to
recruitment in the organization for both the models assuming that (i) the
inter-decision times are exchangeable constantly correlated and (ii) the
threshold distribution is continuous.
The rest of this chapter is organised as follows: In section
5.2, description of Model 1 is given and based upon the aforesaid
univariate policy of recruitment, analytical expressions for mean and
variance of the time to recruitment are obtained. A closed form of these
analytical expressions are obtained by assuming specific distributions.
For a better understanding, the results are numerically illustrated and
relevant conclusions are presented. In section 5.3 Model 2 is described
and a similar computation is carried out using a different univariate
policy of recruitment. Model 2 of this chapter extends the results of
Sathiyamoorthy and Elangovan (1998) for a two graded manpower
system when the inter-decision times are correlated.
5.2 MODEL 1: DESCRIPTION AND ANALYSIS OF THE MODEL
In this section a two graded manpower system is considered and
the description of the model is given below:
82
Assumptions:
1. An organization having two grades (grades A and B) takes
policy decisions at random epochs in [0, ) and at every
decision making epoch, a random number of persons quit
the organization.
2. There is an associated loss of manpower to the organization if a person
quits and it is linear and cumulative.
3. Each grade has its individual independent threshold. If the
total number of persons who leave the organization crosses
the maximum of the two thresholds, recruitment becomes
necessary. In other words, recruitment is made whenever
the cumulative number of exits in the three grades crosses
both the thresholds.
4. Mobility of manpower from one grade to the other grade is
allowed.
Notations:
iU : time between the 1th
i and thi decision epoch. 'iU s are
exchangeable and constantly correlated exponential random variables.
iX :discrete random variable denoting the total number of
persons who leave the organization from the two
83
grades at the thi decision epoch. 1,2,... 'ii X s independent and
identically distributed random variables.
,A BY Y :continuous random variables denoting the threshold levels
for grades A and B respectively and the distribution of AY
and BY is exponential.
Y : max ,A BY Y Y
kV t :probability that there are exactly k decision making epochs
in (0, ]t .
T :a continuous random variable denoting the time for
recruitment in the organization.
L t :cumulative distribution function of T.
.AH :distribution function of AY follows exponential
distribution with the parameter 1
.BH :distribution function of BY follows exponential
distribution with the parameter 2
.H : distribution function of Y.
1a 1
0
: r
i
r
e P X r
84
2a 2
0
: r
i
r
e P X r
3a 1 2
0
:r
i
r
e P X r
and 1 , 1,2,3i ia a i
.kF : cumulative distribution function of 1
k
i
i
U
*: Laplace – Stieltje’s transform.
R : correlation between any iU and jU i j
,n x 1
0
:
x
r ne d
a : mean of , 1,2...iU i
b : 1a R
m
1:
1m m s
bs
E T : Mean time for recruitment
V T : Variance of time to recruitment
5.2.1 Main Results
In this subsection, expressions for , * ,L t L s E T and V T
are obtained.
First we shall obtain the distribution .H x
The distribution functions of AY and BY are given by
11 x
AH x e
85
and
21 x
BH x e
respectively.
Since max , ,A BY Y Y the distribution function H x is
given by
BH x H x H x
1 21 21x xx xH x e e e
(5.2.1)
Next we shall obtain .L t
0k
P T t P
{exactly k decisions in (0, ]t and the threshold
levels are not crossed}
0 1
k
k i
k i
V t P X Y
P T t 0 1
k
k i
k i
V t P X Y
(5.2.2)
Using the law of total probability and (5.2.1)
1
k
i
i
P X Y
1 1 1 1
/k k k k
i i i
i i i i
P Y X X r P X r
0 1
k
i
r i
P Y r P X r
86
1 2
0 1 0 1
k kr r
i i
r i r i
e P X r e P X r
1 2
0 1
kr
i
r i
e P X r
(5.2.3)
Consider
1
0 1
kr
i
r i
e P X r
1
1 1
0 1
k
i
i
k Xr
i
r i
e P X r E e
1 1
1
kX
i
E e
i.e., 1
1
0 1
kr k
i
r i
e P X r a
(5.2.4)
Similarly
2
0 1
kr
i
r i
e P X r
(5.2.5)
1 2
3
0 1
kr k
i
r i
e P X r a
(5.2.6)
Using (5.2.4), (5.2.5) and (5.2.6) in (5.2.3) we get
1
k
i
i
P X Y
1 2 3
k k ka a a (5.2.7)
87
From (5.2.1) and (5.2.7), we get
P T t 1 2 3
0
k k k
k
k
V t a a a
P T t 1 1 2 3
0
k k k
k k
k
F t F t a a a
(5.2.8)
Consider
1
0
k
k k i
k
F t F t a
,
1
0
k
k k i
k
F t F t a
1
0
1 , 1,2,3,..k
k i
k
F t a i
(5.2.9)
Since
1 ,L t P T t using (5.2.8) and (5.2.9) we get
L t 1 1 1
1 1 2 2 3 3
0
k k k
k
k
F t a a a a a a
(5.2.10)
(5.2.10) gives the distribution of T.
Next we find the Laplace-Stieltje’s transform of L t . (i.e.,) * .L s
From Gurland (1955) the cumulative distribution function of
kF x is given by
kF x
10
,
1 ,11
i
ii
xk i
RK bR
k iR kR
(5.2.11)
88
Where 1 2 .... k is the characteristics function of the joint
distribution of any k random variables from 1.k k
X
The Laplace-Stieltje’s transform of kF t is given by
*
kF s 0
st
ks e F t dt
1
1 1 / 1 1k
bs kRbs R bs
(5.2.12)
Taking Laplace-Stieltje’s transform on both sides of (5.2.10)
*L s 1 1 1
1 1 2 2 3 3
1
k k k
k
k
F t a a a a a a
(5.2.13)
Using (5.2.12) in (5.2.13) we get
*L s
1 1 1
1 1 2 2 3 3
1
1 11 1
k k k
k
k k
F t a a a a a a
kRbsbs
R bs
*L s
1 1 1
1 1 2 2 3 3
1 11
1
kk k k
k
ma a a a a a
kR m
R
(5.2.14)
(5.2.14) gives the Laplace-Stieltje’s transform of L t
Where
1
1m
bs
We now obtain E T
89
Since E T 0
* ,s
dL s
ds
From (5.2.14) we get
E T
1 1 1
1 1 2 2 3 3
1
0
11
1
kk k k
k
s
d ma a a a a a
kR mds
R
(5.2.15)
Now for 1,2,3i
11
0
11
1
kk
i i
k
s
d ma a
kR mds
R
1 4
1
21
0
11
1 1
11
1
k
k
i l
k
s
dmkR
kR m dm dskm m
R ds Ra a
kR m
R
(5.2.16)
1
1
11
k
i i
k
Ra a kb
R
11 1
11
kk
i i
k
d ma a
kR mds
R
1
1
1,
1
k
i i
k
a a kbR
Since
,1
ba
R
for 1,2,3i
90
1
1
1
1
k
i i
k
a a kbR
1
1
k
i i
k
a a ka
21 2 3 ...i i ia a a a
i
a
a (5.2.17)
From (5.2.16) and (5.2.17), we get
11 1
11
kk
i i
k
d ma a
kR mds
R
,i
a
a 1,2,3i (5.2.18)
Using (5.2.18) in (5.2.15) we get
E T 1 2 3
1 1 1a
a a a
i.e., E T 1 2 3
1 1 1
1
b
R a a a
(5.2.19)
(5.2.19) gives the mean time for recruitment.
Now, we obtain V T . It is known that
22V T E T E T (5.2.20)
and
2
2
2 0*
s
dE T L s
ds (5.2.21)
Now for 1,2,3i
91
2
1
21 1
11
kk
i i
k
d ma a
kR mds
R
21
21 1 1
kk
i i
k
d ma a
ds kC m
(5.2.22)
Suppose
.1
RC
R
From (5.2.22) one can show that
2
21 1
11
k
k
d m
kR mds
R
1
3 21
21
1 1 1 1
k k
i i
k
dmkC
dm dsa a kC mds kC m kC m
2 21 1
221 1
k k
dmkC
dm d m dm dskm k mds ds ds kC m
2
2
21
2 2 2
11
1 1 1
k
k
dmk m
d mdsm
dskC m kC m
(5.2.23)
Since 2
2
20, 1; ; 2
dm d ms m b b
ds ds from (5.2.23)
2
1
21 1
11
kk
i i
k
d ma a
kR mds
R
92
= 1 2 2 2 2 2 2 2 2 2
1
2 2 1k
i i
k
a a k C b k Cb kCb k Cb k k b
1,2,3i (5.2.24)
Since
1
1
k
i
k
ka
21 2 3 ....a a
and
2 1
1
k
i
k
k a
21 4 9 ....a a
From (5.2.24),
2
1
21 1
11
kk
i i
k
d ma a
kR mds
R
2
2
2
2 1
1i
a
R a
(5.2.25)
Using (5.2.25) in (5.2.21) we get
2E T
22 22
2
1 2 3
2 1 1 1
1
a
R a a a
(5.2.26)
Using (5.2.19) and (5.2.26) in (5.2.20), we get
V T
22 22
2
1 2 3
2 1 1 1
1
a
R a a a
2
2
1 2 3
1 1 1a
a a a
(5.2.27)
Equation (5.2.27) gives the variance of time to recruitment.
93
5.2.2 Special Case
In this section explicit expressions for the mean and variance of
time to recruitment are obtained by assuming specific distributions.
Suppose , 1,2...iX i follows Poisson distribution with
parameter . In this case
1a 1
0
r
i
r
e P X r
1
0 !
rr
r
e er
i.e., 1a 11 e
e
Doing similar computations for 2a and 3a and using in (5.2.19)
and in (5.2.27)we get the following results.
E T 31 2 11 1
1 1 1
1 11 1ee e
b
R ee e
(5.2.28)
and
V T 31 2
2
2 2 2211 1
2 1 1 1
111 1
ee e
a
Ree e
1 2 3
2
21 1 1
1 1 1
11 1e e e
a
ee e
(5.2.29)
Equations (5.2.28) and (5.2.29) gives the explicit expression for
mean and variance of the time to recruitment for the special case.
94
5.2.3 Numerical Illustration
In this section, the analytical expression obtained in (5.2.29) and
(5.2.30) are numerically illustrated and relevant conclusions are made.
Fixing 1 20.5; 0.2; 2a and varying R and the values of
E T and V T are computed and tabulated in Table 5.2.1.
Table 5.2.1
R 1 2 3 4 5
0.5
E T 0.5049 0.3746 0.3441 0.3353 0.3329
V T 0.3944 0.2281 0.1957 0.1869 0.1846
0.4
E T 0.6311 0.4682 0.4301 0.4191 0.4161
V T 0.3943 0.2305 0.1985 0.1897 0.1874
0.2
E T 0.9467 0.7023 0.6452 0.6286 0.6242
V T 0.8872 0.5787 0.4466 0.4268 0.4170
0.5
E T 1.5147 1.1237 1.0323 1.0058 0.9988
V T 3.5497 2.0532 1.7617 1.6818 1.6612
0.8
E T 3.7866 2.8092 2.5807 2.5146 2.4969
V T 61.7530 35.2841 30.1531 28.7490 28.3840
95
5.2.4 Conclusions
From the above table, we make the following observations.
(i) The mean and variance of time to recruitment increases as
decreases, keeping other parameters fixed. In other words when the
average loss of manhours decreases, the mean time for recruitment
increases.
(ii) The mean and variance of time to recruitment
increases as R increases, keeping other parameters fixed.
(iii) The mean and variance of recruitment decreases for negative
correlation and increases for positive correlation when both and R
varies.
5.3 MODEL 2: DESCRIPTION AND ANALYSIS OF THE MODEL
Assumptions:
1. An organization having two grades (grade A and grade B) takes policy
decisions at random epochs in [0, ) and at every decision making
epoch, a random number of persons quit the organization.
2. There is an associated loss of manpower to the organization if a
person quits and it is linear and cumulative.
3. Each grade has its individual independent threshold. If the total
number of persons who leave the organization crosses the minimum of
96
the two thresholds, recruitment becomes necessary. In other words,
recruitment is made whenever the cumulative number of exits in the
two grades crosses any one of the thresholds.
4. Mobility of manpower from one grade to the other grade is not
allowed.
Notations:
iU : Time between the 1th
i and thi decision epoch. 'iU s are
exchangeable and constantly correlated exponential random variables.
iX : Discrete random variable denoting the total number of
persons who leave the organization from the two
grades at the thi decision epoch. 1,2,... 'ii X s independent and
identically distributed random variables.
,A BY Y : continuous random variables denoting the threshold levels
for grades A and B respectively and the distribution of AY and BY is
exponential.
Y : max ,A BY Y Y
kV t : probability that there are exactly k decision making epochs
in (0, ]t .
T :a continuous random variable denoting the time for
recruitment in the organization.
97
L t :cumulative distribution function of T.
.AH :distribution function of AY follows exponential
distribution with the parameter 1 .
.BH :distribution function of BY follows exponential
distribution with the parameter 2 .
.H : distribution function of Y.
1a 2
0
: r
i
r
e P X r
3a 3 3: 1a a
.kF : cumulative distribution function of 1
k
i
i
U
*: Laplace – Stieltje’s transform.
R : correlation between any iU and jU i j
,n x 1
0
:
x
r ne d
.
a : mean of , 1,2...iU i
b : 1a R
m
1:
1m m s
bs
98
E T : mean time for recruitment
V T : variance of time to recruitment
5.3.1 Main Results
In this subsection, expressions for , * ,L t L s E T and V T
are obtained.
The threshold distributions for grade A and grade B are given by
11 x
AH x e
and
21 x
BH x e
Respectively.
Since max , ,A BY Y Y the distribution function H x is
given by
1 2x xH x e e (5.3.1)
Next we shall obtain .L t
0k
P T t P
{k instants of exits in (0, t]} and the cumulative
loss of manpower in these k decisions does not reach the threshold
levels}.
0 1
k
k i
k i
V t P X Y
(5.3.2)
99
Proceeding as in Model l, it is found that
1
k
i
i
P X Y
1 2
3
1
i
kU k
i
E e a
(5.3.3)
1k k kV t F t F t 0,1,2k
From (5.3.2) and (5.3.3) we get
P T t 1 3
0
k
k k
k
F t F t a
(5.3.4)
As in model 1 , * ,L t L s E T and V T can be obtained as
follows:
L t 1
3 3
1
k
k
k
a F t a
(5.3.5)
*L s
13 3
1 11
1
kk
k
ma a
kR m
R
(5.3.6)
E T 3 31
b a
a R a
(5.3.7)
V T
2
2
3
1 21
bR
a R
(5.3.8)
Equations (5.3.7) and (5.3.8) gives the mean and variance of time
to recruitment for Model 2 respectively.
100
5.3.2 Special Case
Assume that , 1,2...iX i follows Poisson distribution with
parameter
In this case, we get
1 2
3
ra E e
1 2
0
r
i
r
e P X r
1 2
0 !
rr
r
e er
i.e.,
1 2
3
e
a e
(5.3.9)
E T
1 2
1e
b
e R
(5.3.10)
and
V T
1 2
2
21 2
1e
bR
e R
(5.3.11)
Equation (5.3.10) and (5.3.11) gives mean and variance of time
to recruitment for this model.
5.3.3 Numerical Illustrations
In this section, the analytical expression obtained in (5.3.10) and
(5.3.11) are numerically illustrated and relevant conclusions are made
101
Fixing 1 20.5; 0.2; 2a and varying R and the values of
E T and V T are computed and tabulated in Table 5.3.1.
Table 5.3.1
R 1 2 3 4 5
0.5
E T 0.7094 0.4740 0.4040 0.3803 0.6383
V T 0.6950 0.3103 0.2287 0.1997 0.1873
0.4
E T 0.8277 0.5530 0.4748 0.4436 0.4296
V T 0.7421 0.3313 0.2442 0.2132 0.2000
0.2
E T 1.2415 0.8296 0.7122 0.6655 0.6445
V T 1.6698 0.7455 0.5495 0.4798 0.4499
0.5
E T 1.6554 1.1061 0.9496 0.8893 0.8593
V T 3.7841 1.6895 1.2453 1.0872 1.0196
0.8
E T 4.9661 3.3183 2.8489 2.6619 2.2778
V T 112.3491 50.1611 36.9729 32.2786 30.2729
5.4 CONCLUSIONS
From the above table, we make the following observations.
(i) The meantime for recruitment increases as decreases, keeping
other parameters fixed. In other words, when the average loss of
manhours decreases, the mean and variance for recruitment increases.
102
(ii) The mean and variance of time to recruitment increases as R
increases, keeping other parameters fixed.
(iii) The mean and variance of recruitment decreases for negative
correlation and increases for positive correlation when both and R
varies.
103
CHAPTER 6
MARKOV ANALYSIS OF BUSINESS WITH TWO LEVELS
AND MANPOWER WITH THREE LEVELS
6.1 INTRODUCTION
Nowadays labour has become a buyers market as well as seller's
market. Any company normally runs on commercial basis wishes to keep
only the optimum level of any resources needed to meet company's
requirement at any time during the course of the business and manpower
is not an exception. This is spelt in the sense that a company does not
want to keep manpower more than what is required. Hence, retrenchment
and recruitment are common and frequent in most of the companies now.
Recruitment is done when the business is busy and shed manpower when
the business is lean. Equally true with the labour, has the option to switch
over to other jobs because of better working condition, better emolument,
proximity to their living place or other reasons. Under such situations the
company may face crisis because business may be there but manpower
may not be available. If skilled labourers and technically qualified
persons leave the business the seriousness is worst felt and the company
has to hire paying heavy price or pay overtime to employees.
In this chapter considered are two characteristics namely
manpower and business. A formula is derived for the steady state rate of
crisis and the steady state probabilities. The situations may be that the
104
manpower may be fully available, insufficiently available or hardly
available, but business may fluctuate between full availability to nil
availability. The steady state probabilities of the continuous Markov
chain describing the transitions in various states are derived and critical
states are identified for presenting the cost analysis. Numerical
illustrations are provided.
6.2 ASSUMPTIONS
1. There are three levels of Manpower namely Manpower is full, is
moderate and Manpower is nil.
2. There are two levels of business namely (I) business is fully
available (2) business is lean or nil.
3. The time T during which the manpower remains continuously
moderate and becomes nil has exponential distribution with
parameter 10. The time R required to complete recruitment for
filling up of vacancies from level nil to moderate level is
exponentially distributed with parameter 01.
4. The time T' during which the Manpower remains continuously full
and becomes nil has exponential distribution with parameter 20 and
the time R' required to complete full recruitment from nil level is
exponentially distributed with parameter 02 .
105
5. The period of time T" during which the Manpower is continuously
full becomes moderate has exponential distribution with parameter
21 and the period of time R" required for recruitment from
insufficient to full is exponentially distributed with parameter 12 .
Random variables T and R; T’ and R; T" and R" are all independent.
6. The busy and lean periods of the business are exponentially
distributed with parameters ‘a’ and ‘b’ respectively.
6.3 SYSTEM ANALYSIS
The Stochastic Process X t describing the state of the system is a
continuous time Markov chain with 4 points state space as given below in
the order of Manpower and Business
0 0 , 0 1 , 1 0 , 1 1 , 2 0 , 2 1S (6.3.1)
where
2 -Refers to full availability in the case of manpower
1 -Refers to semi availability or insufficiently available manpower
and it refers to busy period in the case of business.
0 -Refers to shortage/lean/non availability manpower or business.
The infinitesimal generator Q of the continuous time Markov
chain of the state space is given below which is a matrix of order 4.
106
Q =
MP/B (0 0) (0 1) (1 0) (1 1) (2 0) (2 1)
(6.3.2)
(0 0) 1 b 01 0 02 0
(0 1) a 2 0 01 0 02
(1 0) 10 0 3 b 12 0
(11) 0 10 a 4 0 12
(2 0) 20 0 21 0 5 b
(2 1) 0 20 0 21 a 4
1 02 02 2 02 02 2 10 12= ( ) ( ) ( ) , , ,b a b
4 10 12 5 20 21 4 21 21, ,( ) .( ) ( )a b a (6.3.3)
The occurrences of transitions in both manpower and business are
independent, the individual infinitesimal generator of them are given by:
1. The infinitesimal generator of business is given below by a matrix of
order
B =
B 0 1
0 b b
1 a a
and the steady state probabilities are
0B
a
a b
and 1 .B
b
a b
2. The infinitesimal generator of manpower is given below by the matrix
of order 3,
107
M =
M 0 1 2
0 01 02 01 02
1 10 10 12 12
2 20 21 21 20
The steady state probabilities of manpower are:
00
0 1 2
,M
d
d d d
1
1
0 1 2
,M
d
d d d
2
2
0 1 2
M
d
d d d
Where
0 20 12 20 10 21 10d
1 20 12 20 10 21 10d
2 10 02 12 02 12 01d
The steady state probability vector of the matrix Q can be derived
easily by using
0Q and 1e
000 2
0
,
i
ad
Z d
0
01 2
0
,
i
bd
Z d
1
10 2
0
,
i
ad
Z d
111 2
0
,
i
bd
Z d
0
20 2
0
,
i
ad
Z d
1
21 2
0
,
i
ad
Z d
(6.3.4)
Where 0 1
2
0
2 i d d d Zd a b
108
When the business is available, either full manpower or moderate
manpower must be available. When it is not so this will create heavy loss,
we shall call this situation as crisis.
The crisis state is {(0 1)} and the crises occur when there is full
business but manpower is NIL.
Now the rate of crisis in steady state (C) is obtained as follows.
P(crisis in[ ]tt t )
0 1 / 0 0 0 0P X t t X t P X t
0 1 / 2 1 2 1P X t t X t P X t
0 1 / 11 11 .P X t t X t P X t O t
Taking limit as 0,t
00 20 21 10 11tC bP t P t P t
00 20 21 10 11[ ] 1 tC t bP t P t P t
that is
00 10 21 10 11[ ]C b
Using the steady state probabilities, obtained
0 10 0 10 1
bC ad d d
ZY (6.3.5)
Where Z a b and 0 1 2 .Y d d d
109
6.4 NUMERICAL ILLUSTRATION
Now taking the values of the parameters in the model as below, can
find the steady probabilities and the rate of crises using the formulas
(6.3.4) and (6.3.5) respectively.
10 = 4, 21 = 5, 20 = 2, 12 = 8, 01 = 4, 02 = 7, a = 8 and b = 9
Steady state
probability Value
00 0.0818
01 0.0922
10 0.1432
11 0.1411
20 0.2455
21 0.2742
Total 1.0000
Now assigning the values 9, 12, 15, 17b and 19 we calculate the
corresponding rate of crisis and are given below in the table:
b Y
9 1.9355
12 2.1913
15 2.3819
17 2.4865
19 2.5701
110
The graph of the steady state crisis is given below taking the values
of b on the x-axis and the value of C∞ on y axis.
Y Values
0
0.5
1
1.5
2
2.5
3
1 2 3 4 5
Y
We find that as the value of parameter b increases the crisis
rate also increases. Also we observe that the cost of doing business
is very heavy if the manpower is full but there is no business. Under
circumstances we should fetch business at premium rate or offer
heavy discount to get business. The cost of business is comparatively
low when the business is full and the manpower is also full. The
same holds in the case of manpower is moderate whereas the
business may be dull or busy.
6.5 CONCLUSION
Though there may be many factors (characteristics) affecting a
business, the most vital among all are manpower and money. If these are
kept strong all other factors can be managed. Finance must very
meticulously handled by experienced person with specialization. Charted
111
accountants, persons with company secretary ship qualification, cost
accountants, Income tax specialist, sales tax specialist, etc must be
employed and salary should not stand as hurdles to employ such
specialist. Skilled labourers must be carefully handled. They are assets to
the company, their genuine demands must be met but at the same time
optimal manpower should only be maintained, this must be done in the
interest of the company and keeping in mind the welfare of the
employees. There can be retrenchments, but must be carried out in a wise
way as may not affect the morale of the employees. Proper training must
be given to the employees.
Every chapter is special in its own way. A business may be
governed by environments but by comparing their strength, we can
always reduce to two characteristics which may bring in a crisis state. A
crisis state is one where business gets affected because of shortage
occurring in the characteristics governing the business. Application of
game theory can reduce the number of environments to two. The
application of the formulas in the models will give the rate of crisis. Now
a business concern can take precautionary measures to avoid coming of
such situations. Cost at different situations can be worked out using the
formulas in the models. Necessary steps may be taken to avoid incurring
heavy expenditure. A manufacturing concern depends on machines for its
production.
112
The failure of machineries will result in heavy expenditure for
production. Such situations are dealt deriving formulas to determine cost
and the rate of crisis. This model is sure to have its application for a
manufacturing concern. Just like machines the computers have
importance in any business concern and have become totally
indispensable. A model dealing with its application gives formulas and is
a very useful to avoid a crisis state occurring due to software or hardware
failure. Instead of two levels (0 for nil availability and 1 for full
availability) three levels of manpower is dealt. Models with this new
concept are useful because we find many business concerns run the
business with inadequate or insufficient staff. Formulas will help to over
come crisis states and avoid incurring heavy costs because of manpower.
113
CHAPTER - 7
EXPECTED TIME FOR RECRUITMENT IN A TWO
GRADEDMANPOWER SYSTEM ASSOCIATED WITH
CORRELATEDINTER-DECISION TIMES WHEN THRESHOLD
DISTRIBUTION HAS SCBZ PROPERTY
7.1 INTRODUCTION
For a single graded system, Sathiyamoorthy and Elangovan
(1998(a)) have obtained the mean and variance of the time to recruitment.
(i) When the number of exits forms a sequence of independent and
identically distributed random variables,
(ii) The random threshold is geometric and (iii) the inter-decision
times are independent and identically distributed random variables. Later,
for the same manpower system, Sathiyamoorthy and Parthasarathy (2003)
have obtained the mean and variance of the time to recruitment when
(i) The loss of manhours process isa sequence of independent and
identically distributed random variables and (ii) the random threshold has
Setting the Clock Back to Zero (SCBZ) property.
Mariappan and Srinivasan (2001(a)) have also obtained the
meantime for recruitment in a single graded system using shock model
approach when the inter-decision times are correlated exchangeable and
exponential random variables also they have obtained the mean time for
recruitment in a single graded system using shock model approach when
114
the bi-variate process formed by loss of manpower and inter-decision
times forms a correlated renewal sequence.
In this chapter, an organization with two grades subjected to loss of
manpower due to staff depletions caused by policy decisions taken by the
organization is considered. Assuming that each grade has its own random
threshold whose distribution has SCBZ property and the inter-decision
times are exchangeable and constantly correlated exponential random
variables, two mathematical models are constructed based upon an
appropriate univariate policy of recruitment. The objective of this chapter
is to find the mean time for recruitment in the organization for both the
models.
The rest of this chapter is organised as follows: In Model 1 is
given, analytical expression for mean time to recruitment is obtained and
the special cases are discussed.
In Model 2 is described and a similar computation is carried out
using a different univariate policy of recruitment. In section 4.4, both the
models are numerically illustrated and relevant conclusions are made.
7.2 MODEL 1: DESCRIPTION AND ANALYSIS OF THE MODEL
Assumptions
1. An organization having two grades (grades A and B) takes policy
decisions at random epochs in 0, and at every decision making
epoch, a random number of persons quit the organization.
115
2. There is an associated loss of man hours to the organization if a
person quits and it is linear and cumulative.
3. Each grade has its own threshold level and the
thresholddistribution has SCBZ property. Recruitment is
madewhenever the total number of exits exceeds the threshold
level in both the grades.
4. The inter–decision times are exchangeable constantly correlated
exponential random variables.
5. The process which generates the number of exits and the threshold
are mutually independent.
6. Mobility of man power from one grade to the other is permitted.
Notations
iU : time between the 1th
i and thi decision epoch. ’iU s are
exchangeable and constantly correlated exponential random variables .
iX : discrete random variable denoting the total number of persons
who leave the organization from the two grades at the thi decision
epoch. 11,2,..... 'i X S independent and identically distributed random
variable
,A BY Y : continuous random variables denoting the threshold of
levels for the grades A and B respectively and the distributions AY and
BY follows SCBZ property.
116
,(: )max ,A BY Y Y Y
kV t : probability that there are k decisions in (0,t].
.g : probability density function of , 1,2, ..iX i
)*(.g : Laplace transform of .g
.kg : k -fold convolution of .g
.f : probability density function of inter-decision times
.kf : k-fold convolution of .f
.kF : k-fold convolution of .F
.AH : distribution function of AY with parameters 1 2, and 1
.BH : distribution function of YB with parameters 3 4, and
2
.H : distribution function of Y.
T : time for recruitment in the organization
L t : distribution function of T
*L s : Laplace-Stieltje’s transform of L t .
R: correlation between any iU and ,jU i j
1
0
, :
x
nn x e d
a : mean of , 1,2......iU i
b : 1a R
117
m :
1
1m m s
bs
E T : mean time for recruitment .
7.2.1 Main Results
In this subsection an analytical expression for the mean time to
recruitment is obtained.
Since Y= max ,( ),A BY Y , the distribution of Y is given by,
A BH x H x H x
1 1 A BH x H x H x
The probability distribution of the thresholds YA and YB for the two
grades respectively are given by,
1 1 2
1 11x x
AH x p e q e
And
3 2 4
2 21x x
BH x p e q e
where
1 2 11 1
1 1 2 1 1 2
; ;p q
3 4 22 2
2 3 4 2 3 4
;p q
Where 1 2and are the parameters of the exponentially distributed
truncated random variables.
Since max , ,A BY Y Y
118
1 1 3 3 1 2
1 21x x
H x p e p e
2 4 2 4 1 3 1 2
1 2 1 2 1 2
x x x rq e q e q q e p p e
3 4 1 2 3 2 2 4
1 2 1 2 1 2
x x xp q e q p e q q e
(7.2.1)
The probability that the threshold level is not reached till ‘t’
is, 0k
P T t P
{ k instants of exits in (0,t] }and the cumulative loss of
manpower in these k decisions does not reach the threshold level}
0 1
k
k
k i
V t P X Y
(7.2.2)
We now calculate 1
.k
i
P X Y
Using the law of total probability and (7.2.1),
1 0 0 0 0
|k
i i
i r k k k
P X Y P Y X X r P X Y
0 0
i
r k
P Y r P X r
1 1 3 2
1 2
0 0 0 0
r r
i i
r k r k
p e p X r p e X r
2 4
1 2
0 0 0 0
r r
i i
r k r k
q e p X r q e p X r
1 3 1 2 1 4 1
1 2 1 2
0 0 0 0
r r
i i
r k r k
p p e p X r p q e p X r
119
2 3 2 2 4
2 1 1 2
0 0 0 0
r r
i i
r k r k
p q e p X r q q e p X r
(7.2.3)
write
1 1 3 2
1 2
0 0
; ;r r
i i
r r
a e p X r a e p X r
1 4
3 4
0 0
; ;r r
i i
r r
a e p X r a e p X r
1 3 1 2 1 3 1
5 6
0 0
; ;r r
i i
r r
a e p X r a e p X r
1 3 1 2 1 3 1
5 6
0 0
; ;r r
i i
r r
a e p X r a e p X r
Now
0
, 1,2,.......ixr
i
r
e p X r E e i
and
0 1 1
i
kXr
i
r i i
e p X r E e
1 1 1 1
1 1
0 0 1 0
kr
i i
r k i k
p e p X r p E e X
1 1
1 1 0
1 1
0 0 1
i
k
k Xr
i
r k i
p e p X r p E e
= 1 1
kp a (7.2.5)
Similarly for other terms of (7.2.3) we can show that
120
3 2
2 2 2
0 0
r k
i
r k
p e p X r p a
(7.2.6)
2
1 1 3
0 0
r k
i
r k
q e p X r q a
(7.2.7)
4
2 2 4
0 0
r k
i
r k
q e p X r q a
(7.2.8)
1 3 1 2
1 2 1 2 5
0 0
r k
i
r k
p p e p X r p p a
(7.2.9)
1 4 1
1 2 1 2 6
0 0
r k
i
r k
p q e p X r p q a
(7.2.10)
2 3 2
2 1 2 1 7
0 0
r k
i
r k
p q e p X r p q a
(7.2.11)
2 4
2 41 2 1 2 1 2 8
0 0
k
r k
i
r k
q q e p X r q q d q q a
(7.2.12)
Using the results (7.2.5) to (7.2.12) in (7.2.3)
1 1 2 2 1 3 2 4
0 1 2 5 1 2 6 2 1 7 1 2 8
k k k k
k k k k kk
p a p a q a q ap T t V t
p p a p q a p q a q q a
Now L*(s) can be obtained as follows
1 1 2 2 1 3 2 4
1
0 1 2 5 1 2 6 2 1 7 1 2 8
k k k k
k k k k k kk
p a p a q a q ap T t F t F t
p p a p q a p q a q q a
(7.2.13)
Since 1L t p T t p T t
121
1 1 1 2 2 2 1 3 3 2 4 4
0 1 2 5 5 1 2 6 6 2 1 7 7 1 2 8 8
k k k k
k k k k kk
p a a p a a q a a q a aL t F t
p p a a p q a a p q a a q q a a
(7.2.14)
From Gurland, J (1955), Laplace - Stieltje’s transform of kF t is
given by
1*
1 11 1
k
k
F skRbs
bsR bs
Taking Laplace-Stieltjes transform on both sides of (7.2.14), we get
1
1*
11 1
k
k
bsL s X
kRbs
R bs
1 1 1 2 2 2 1 3 3 2 4 4
1 2 5 5 1 2 6 6 2 1 7 7 1 2 8 8
k k k k
k k k k
p a a p a a q a a q a a
p p a a p q a a p q a a q q a a
*L s = 1 1
11
k
k
m
kR m
R
1 1 1 2 2 2 1 3 3 2 4 4
1 2 5 5 1 2 6 6 2 1 7 7 1 2 8 8
k k k k
k k k k
p a a p a a q a a q a a
p p a a p q a a p q a a q q a a
(7.2.15)
Since 0
*s
dE T L s
ds (7.2.16)
Using (7.2.15) in (7.2.16) the mean time for recruitment is found to be
122
1 2 1 2 1 2 1 2 2 1 1 2
1 2 3 4 5 6 7 81
b p p q q p p p q p q q qE T
R a a a a a a a a
(i.e) 1 2 1 2 1 2 1 2 2 1 1 2
1 2 3 4 5 6 7 8
p p q q p p p q p q q qE T a
a a a a a a a a
(7.2.17)
Where , 1,2......ia i and 1i ia a are given by (7.2.4),(7.2.17)
gives the mean time for recruitment.
7.2.2 Special Case
Suppose , 1,2...iX i follows Poisson distribution with parameter
1 1
1
0
r
i
r
a e p X r
1 1
0
.
!
rr
r
ee
r
1 1
0 !
r
r
ee
r
1 1.ee e
1 11
1
e
a e
(7.2.18)
Similarly,
3 1 1 3 1 22 41 11 1
2 3 4 5; ; ;e ee e
a e a e a e a e
3 2 21 4 1 2 411 1
6 7 8; ;ee e
a e a e a e
(7.2.19)
Using (7.2.18) and (7.2.19) in (7.2.17) we get
123
1 1 23 2
4 1 4 11 3 1 2
2 43 2 2
1 2 1
1 11
2 1 2 1 2
1 11
2 1 1 2
11
1 11
1 11
11
e ee
e ee
ee
p P q
e ee
q p q p qE T a
e ee
p q q q
ee
(7.2.20)
Equation (7.2.20) gives the meantime for recruitment for the
special case.
7.2.3 Numerical Illustration
In this section model 1 is numerically illustrated and relevant
conclusions are made.
Fixing 1 2 1 2 3 42; 0.8; 0.5; 0.4; 0.3; 0.6; 0.2B and
varying R and , the values of E T are computed and tabulated in table
7.2.1.
Table 7.2.1
R
1 2 3 4 5
0.4 0.9962 0.5307 0.3748 0.2967 0.2503
0.2 1.1622 0.6192 0.4372 0.3462 0.2920
0.2 1.7434 0.9288 0.6558 0.5192 0.4380
0.4 2.3245 1.2384 0.8744 0.6923 0.5840
0.8 6.9734 3.7151 2.6233 2.0769 1.7521
124
7.2.4 Conclusion
From the above table, we make the following observations.
(i) The mean time for recruitment increases as decreases,
keeping other parameters fixed. In other words, when the number
of exits decreases on the average, the mean time for recruitment
increases.
(ii) The mean time for recruitment increases as R increases,
keeping other parameters fixed.
(iii) The meantime to the recruitment decreases for negative
correlation and increases for positive correlation when R and
varies simultaneously.
7.3 Model 2: Description and Analysis of the Model
Assumptions
1. An organization having two grades A and B takes policy decisions
at random epochs in 0, and at every decision making epoch, a
random number of persons quit the organization.
2. There is an associated loss of man hours to the organization if a
person quits and it is linear and cumulative.
3. Each grade has its own threshold level and the threshold
distribution has SCBZ property. Recruitment is made whenever the
total number of exits exceeds in any one of the two threshold.
125
4. The inter – decision times are exchangeable and constantly
correlated.
5. The process which generates the number of exits and the threshold
are mutually independent.
6. Mobility of man power from one grade to the other is permitted.
Notations
iU : time between the 1th
i and thi decision epoch. ’iU s are
exchangeable and constantly correlated exponential random variables .
iX : discrete random variable denoting the total number of persons
who leave the organization from the two grades at the thi decision
epoch.11,2,..... 'i X s independent and identically distributed random
variable
,A BY Y : continuous random variables denoting the threshold of
levels for the grades A and B respectively and the distributions AY and
BY follows SCBZ property.
,(: )min ,A BY Y Y Y
kV t : probability that there are k decisions in (0,t].
.g : probability density function of X
)*(.g : Laplace transform of .g
.kg : k -fold convolution of .g
126
.f : probability density function of inter-decision times
.kf : k-fold convolution of .f
.kF : k-fold convolution of .F
.AH : distribution function of AY with parameters 1 2, and
1 .BH : distribution function of BY with parameters 3 4,
and 2 .H : distribution function of Y.
T: time for recruitment in the organization
L t : distribution function of T
*L s : Laplace-Stieltje’s transform of L t .
R: correlation between any iU and ,jU i j
1
0
, :
x
nn x e d
a : mean of , 1,2......iU i
b : 1a R
m :
1
1m m s
bs
E T : mean time for recruitment .
7.3.1 Main Results
In this subsection an analytical expression for the mean time to
recruitment is obtained.
As in chapter 2, the distribution of Y is given by,
127
1 H x 1 3 1 2 3 4 1
1 2 1 2
x xp p e p q e
2 3 2 2 4
1 2 1 2
x xq p e q q e
(7.3.1)
As in model 1, it can be show that
L t 1 2 5 1 2 6 2 1 7 1 2 8
1
k k k k
k
k
F t p p a p q a p q a q q a
(7.3.2)
and
*L s
11 2 5 1 2 6 2 1 7 1 2 8
1 /1 / 1 1k
k k k kk
bs kRbs R bs
p p a p q a p q a q q a
1 2 5 5 1 2 6 6
1 2 1 7 7 1 2 8 8
1
11 1
k k k
k kk
p p a a p q a abs
kRbs p q a a q q a a
R bs
*L s
1 2 5 5 1 2 6 6
1 2 1 7 7 1 2 8 81
11
k kk
k kk
p p a a p q a am
kR m p q a a q q a a
R
(7.3.3)
Where 1
1m
bs
Since E T 0
*s
dL s
ds from (7.3.4) one can found that
E T 1 2 1 2 2 1 1 2
5 6 7 81
b p p p q p q q q
R a a a a
i.e., E T 1 2 1 2 2 1 1 2
5 6 7 8
p p p q p q q qa
a a a a
128
Where , 5,6,7,8ia i and 1i ia a are given by (7.2.4)
Equation (7.3.4) gives the mean time for requirement.
7.3.2 Special Case
Suppose , 1,2,..iX i follows Poisson distribution with parameter .
5a 1 3 1 2
0
r
i
r
e P X r
1 3 1 2
0 !
rr
r
ee
r
1 3 1 2e e
i.e., 5a 1 3 1 21 e
e
Similarly,
1 4 11
6 ;e
a e
3 2 21
7 ;e
a e
2 41
8
e
a e
Now,
E T 2 41 3 1 2 1 4 3 21 2
1 2 1 2 2 1 1 2
11 1 1 ee e e
p p p q p q q qa
ee e e
Equation (4.3.7) is an analytical expression to the mean time for
recruitment. In this section model 2 is numerically illustrated and relevant
conclusions are made.
Fixing 1 2 1 2 3 42; 0.8; 0.5; 0.4; 0.3; 0.6; 0.2b
and varying R and , the values of E T are computed and tabulated in
Table 7.3.1.
129
T
R
1 2 3 4 5
0.4 0.3246 0.2079 0.1729 0.1581 0.1508
0.2 0.3787 0.2425 0.2018 0.184 0.1760
0.2 0.5681 0.3638 0.3026 0.2766 0.2640
0.4 0.7574 0.4850 0.4035 0.3688 0.3520
0.8 2.2723 1.4551 1.2106 1.1065 1.0559
7.4 CONCLUSION
From the above table, we make the following observations.
(i) The mean time for recruitment increases as decreases, keeping
other parameters fixed. In otherwords, when the number of exits
decreases on the average, the mean time for recruitment increases.
(ii) The mean time to recruitment increases as R increases, keeping
other parameters fixed.
(iii) The meantime to the recruitment decreases for negative correlation
and increases for positive correlation when R and varies
simultaneously.
112
130
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